DECISION MAKING FOR BRIDGE STOCK...
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1 University of Trento Emiliano Debiasi DECISION MAKING FOR BRIDGE STOCK MANAGEMENT Tutor: Dr. Daniele Zonta 2014
Transcript of DECISION MAKING FOR BRIDGE STOCK...
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University of Trento
Emiliano Debiasi
DECISION MAKING FOR BRIDGE STOCK
MANAGEMENT
Tutor: Dr. Daniele Zonta
2014
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UNIVERSITY OF TRENTO
Civil and Mechanical Structural Systems Engineering
XXVI Cycle
Final Examination 10/04/2014
Board of Examiners
Prof. Maurizio Piazza (Università di Trento)
Prof. Andrea Prota (Università di Napoli Federico II)
Prof. Carmelo Gentile (Politecnico di Milano)
Prof. Helmut Wenzel (Universität für Bodenkultur Wien)
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SUMMARY-SOMMARIO
Bridges in service in most Western Countries were built according to codes with
design loads that are now inconsistent with today’s traffic demands. Currently,
transportation agencies do not know how to respond to transit applications on
their bridges. This thesis focuses on the legal issues entailed by
overweight/oversize load permits issued by transportation agencies. Indeed,
correct decision-making should consider the legal liabilities involved in possible
catastrophic events. In this thesis I illustrate how this problem is guided by the
Department of Transportation of the Italian Autonomous Province of Trento
(APT’s DoT), a medium-sized agency managing approximately one thousand
bridges across its territory. In the basic approach, it does not authorize movement
of overweight loads unless it is demonstrated that their effect is less than that of
the nominal design load. When this condition is not satisfied, a formal evaluation
is carried out in an attempt to assess a higher load rating for the bridge. If, after
the reassessment, the rating is still insufficient, the bridge is classified as sub-
standard and a formal evaluation of the operational risk is performed to define a
priority ranking for future reinforcement or replacement.
Italian version
I ponti in servizio nella maggior parte dei Paesi occidentali, sono stati costruiti in
accordo a normative che prevedevano carichi di progetto che al giorno d’oggi
sono inconsistenti con l’attuale domanda di traffico. Oggigiorno, le agenzie di
trasporto non sanno come comportarsi nel caso di richieste di transito di carichi
eccezionali sui ponti di loro proprietà. Questa tesi si concentra sulle questioni
giuridiche derivanti dai permessi dei carichi eccezionali emessi dalle agenzie di
trasporto. Infatti, un corretto processo decisionale dovrebbe considerare le
responsabilità giuridiche a seguito di eventuali eventi catastrofici. In questa tesi si
illustra come questo problema è stato indirizzato nel caso del Dipartimento dei
Trasporti della Provincia Autonoma di Trento (PAT), un'agenzia di medie
dimensioni che gestisce circa un migliaio di ponti sul suo territorio. Nell'approccio
di base, non si autorizza il transito del carico eccezionale sul ponte a meno che la
sollecitazione non sia inferiore a quella prodotta dal carico nominale di progetto.
Quando questa condizione non è soddisfatta, viene effettuata una valutazione
formale nel tentativo di valutare un carico più elevato per il ponte. Se, dopo
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questa nuova valutazione il risultato è ancora negativo, il ponte viene classificato
come non conforme e viene eseguita una valutazione formale del rischio, allo
scopo di definire una classifica di priorità per il suo futuro rinforzo o sostituzione.
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DEDICATION
A Elena
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ACKNOWLEDGEMENTS
This research was made possible thanks to the financial support of the
Department of Public Works of the Autonomous Province of Trento: Stefano De
Vigili, Luciano Martorano, Guido Benedetti, Matteo Pravda, Rosaria Fontana, Italo
Artico, and Filiberto Bolego are particularly acknowledged. I also wish to thank
Alessandro Lastei, Luca Mattiuzzi, Francesco Chesini, Francesco Demozzi,
Marco Tallandini, Alessandro Lanaro, Davide Trapani and Carlo Cappello all
collaborating in the research group at the University of Trento, for their
contribution to the development of this project.
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CONTENTS
SUMMARY-SOMMARIO ......................................................................................... 3
DEDICATION .......................................................................................................... 5
ACKNOWLEDGEMENTS........................................................................................ 7
CONTENTS ............................................................................................................. 9
1 INTRODUCTION ............................................................................................11
1.1 Motivation/Problem statement ...............................................................11
1.2 Objectives ..............................................................................................13
1.3 Method ...................................................................................................14
1.4 Outline ...................................................................................................15
1.5 Limitations..............................................................................................15
2 PROBLEM STATEMENT ...............................................................................17
2.1 Introduction ............................................................................................17
2.2 State of the art .......................................................................................17
2.3 Current procedure .................................................................................21
2.4 Effect of vehicle velocity ........................................................................23
2.5 Travel conditions ...................................................................................24
2.6 APT overweight vehicle models ............................................................25
2.7 Conclusions ...........................................................................................27
3 CRITERIA FOR RE-ASSESSING BRIDGES FOR OVERWEIGHT LOADS .29
3.1 Introduction ............................................................................................29
3.2 Assessment principle .............................................................................29
3.3 Levels of assessment ............................................................................30
3.4 Procedures for assessment ...................................................................32
3.4.1 Girder bridges ....................................................................................32
3.4.2 Arch bridges.......................................................................................40
3.5 Lack of capacity of substandard bridges ...............................................44
3.6 Risk interpretation of parameter .........................................................50
3.7 Conclusions ...........................................................................................54
4 APPLICATION TO CASE STUDIES ..............................................................57
4.1 Introduction ............................................................................................57
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4.2 Case studies ..........................................................................................57
4.2.1 Fiume Adige.......................................................................................58
4.2.2 Rio Cavallo ........................................................................................75
4.3 Discussion of the results ........................................................................84
4.4 Conclusions ...........................................................................................86
5 DECISION MAKING .......................................................................................87
5.1 Introduction ............................................................................................87
5.2 Basic concepts ......................................................................................88
5.3 Expected utility theory ...........................................................................89
5.4 Decision tree ..........................................................................................90
5.5 Optimization of alpha thresholds .........................................................100
5.6 Bayesian updating of parameters ........................................................103
5.7 Conclusions .........................................................................................107
6 APPLICATION TO THE APT BRIDGE STOCK ...........................................109
6.1 Introduction ..........................................................................................109
6.2 Bridge Management System ...............................................................109
6.3 Strategic objective ...............................................................................117
6.4 Level 0 assessment results .................................................................119
6.5 Decision tree ........................................................................................125
6.6 Prediction of re-assessment results and costs ....................................127
6.7 Application to 20 APT bridges .............................................................131
6.8 Bayesian updating and thresholds of optimization ...........................138
6.9 Conclusions .........................................................................................142
7 CONCLUSIONS AND FURTHER WORK ....................................................143
7.1 Conclusions .........................................................................................143
7.2 Further work ........................................................................................144
REFERENCES ....................................................................................................145
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1 INTRODUCTION
1.1 Motivation/Problem statement
Managing large infrastructure systems is a multi-disciplinary activity requiring
expertise from many areas, including fields of research beyond the typical scope
of the structural engineer. The process of risk analysis and decision-making itself
involves knowledge of the cognitive mechanisms and the psychology of economic
decisions. Mere structural reliability is just one of many aspects affecting
decisions, while economic, social, ethical and legal issues must also be
considered in a risk model. In recent years, transportation agencies have focused
on the problem of overweight traffic management, due to the increase in the
nominal load of heavy vehicles and to the increasing age and deterioration of the
infrastructure.
In fact, most bridges operating today in Europe were built to old codes that
foresaw design loads inconsistent with today’s traffic demand, and currently
transportation agencies do not know how to behave in the case of applications for
transit on their bridges.
A formal re-assessment of old bridges with respect to new design codes would
require analysis of design documents that may not be available and also
structural recalculation; and very often expensive load tests. Because of the
number of old bridges, an agency cannot normally carry the overall cost of a full
analysis and will seek for simplified approaches.
In this thesis I illustrate how this problem can be managed, applying the
methodology to the stock of the Department of Transportation of the Italian
Autonomous Province of Trento (APT’s DoT). Currently, the APT manages
approximately 2410 kilometers of roadways and 953 bridges (Fig. 1).
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Fig. 1: APT bridge stock
With reference to type and year of construction, the APT’s bridge stock has
characteristics similar to those found elsewhere in Europe: most of the bridges
were built after the Second World War, peaking in the 70's (Yue et al., 2010) (Fig.
2). Approximately 66% of the bridges were built of reinforced or prestressed
concrete, 25% have an arch structure and 9% were built in steel or are composite
steel-concrete structures (Bortot et al., 2009) (Fig. 2).
Fig. 2: a) age distribution and b) typological distribution in the APT (Zonta et al., 2007)
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In order to allow effective planning of maintenance policies and to meet the new
requirements, the Department of Transportation of the APT has begun to
collaborate with the Department of Mechanical and Structural Engineering,
University of Trento, to develop a comprehensive Bridge Management System
(APT-BMS) (Bortot et al., 2009) (Fig. 3).
Fig. 3: Web application of APT-BMS (www.bms.provincia.tn.it)
1.2 Objectives
In this thesis I illustrate how the problem of applications for transit of overweight
loads can be guided. In particular, this thesis focuses on the legal issues entailed
by overweight/oversize load permits issued by transportation agencies. Indeed,
correct decision-making should consider the legal liabilities involved in possible
catastrophic events. The main objective is therefore to develop a decision support
system for the management of the transit of the overweight traffic loads on
bridges, in particular for the APT stock. This thesis provides simple, practical rules
for deciding whether an overweight load permit can be issued or not, and if so,
under what restrictions.
Moreover, I apply utility theory to optimize the expected cost in order to propose a
process to guide the APT in making the most cost effective decision when
selecting the appropriate level of verification.
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1.3 Method
In general, permits for overweight vehicles are issued by transportation agencies
based on the total gross weight, the maximum axle load, the distribution of axle
weights and on the axle spacing (see for example Federal Highway
Administration, (2012)).
One of the main problems is the high variability of axle loads and spacing. A set
of predefined overweight load configurations is therefore defined that are more
conservative than other axle combinations having the same gross weight. In the
basic approach, it does not authorize the transit of the overweight load unless it is
demonstrated that the effect is less than that of the nominal design load. It is
worth emphasizing that this approach protects transportation agencies against
legal liabilities in the case of a catastrophic event, because it has authorized the
transit of a vehicle whose effects are less than those produced under the design
assumptions. When the bridge is not verified, a formal evaluation is carried out in
an attempt to assess a higher load carrying capacity for the bridge. If, after the
reassessment, the capacity is insufficient, the bridge is classified as sub-standard
and a formal evaluation of the operational risk is performed in order to define a
priority ranking for future reinforcement or replacement.
The method is proposed as an alternative to the recent methods developed in
order to solve the problem of transit permits. In fact, the permit issuing process
would require a detailed structural analysis, which in turn would require very
laborious analyses. Furthermore, in many cases, the geometrical data necessary
to apply the analyses are not fully available. Recently, to overcome this difficulty
three methods were discussed (Vigh and Kollar, 2006). The three methods are
the following (Vigh and Kollar, 2006):
1) application of a simplified structural analysis with a comparison of the
effects (bending moment, shear force) caused by the design loads and
overweight loads (e.g. Correia and Branco, 2006);
2) comparison of the axle loads using the bridge gross weight formula
(Bridge formula weights, 1994), or their improvements (James et al. 1986;
Chou et al. 1999; Kurt 2000);
3) application of approximate transversal load distributions (Vigh and Kollar,
2007).
The three methods are discussed in detail in Section 2.2.
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1.4 Outline
In this thesis I illustrate how to guide the legal issues arising from the issue of
transit permits by the transportation agencies. In the next Chapter I first introduce
the overweight vehicle permit problem and describe how it is currently formally
managed in the APT. I then defined the travel conditions and a set of predefined
overweight load configurations that are more conservative than other axle
combinations having the same gross weight.
In Chapter 3, I show the multi-level assessment protocol conceived to estimate
the response of the bridge stock to overweight loads. In particular, three levels of
refinement of bridge assessment are foreseen to verify the bridge under
overweight loads. At the end an interpretation of the risk is proposed.
Chapter 4 applies the methodology to two case studies in order to test the
method. The sub-standard girder bridge Fiume Adige and arch bridge Rio Cavallo
are chosen to perform the levels of refinement.
In Chapter 5 I propose a decision making process to guide transportation
agencies in making the most cost effective decision. The utility theory is used to
optimize the expected cost. In addition, a methodology based on Bayesian
statistics is used to update parameters on the basis of a number of case studies.
Then, in Chapter 6 I apply the decision making process to the APT bridge stock in
accordance with the strategic objectives to assess/retrofit the APT bridge stock.
Based on the decision tree proposed, a prediction of the future demand for re-
assessment and costs is presented. Finally I illustrate the optimization of the
thresholds of the lack of capacity based on the results of 20 APT bridge case
studies.
Some concluding remarks are made in Chapter 7.
1.5 Limitations
Although the methodology proposed in this thesis can be used by any
transportation agency, the set of APT vehicle models and the decision tree
proposed (see Section 2.6) have been created following the needs of the APT
and the limitations of the Italian codes. Therefore, the reference loads must be
changed when the DoT needs or the codes are different. However, the needs of
the APT could be similar to many other transportation agencies.
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2 PROBLEM STATEMENT
2.1 Introduction
In recent years, the APT’s DoT has focused on the problems arising from the
increase in the nominal load of heavy vehicles and the increasing age and
deterioration of the infrastructure. A formal re-assessment of old bridges with
respect to new design codes would require analysis of the original design
documents, often unavailable, and structural recalculation; and very often
expensive load tests. Because of the number of old bridges, an agency cannot
normally carry these costs and will seek simplified approaches.
To re-assess the bridges taking into account the current overweight vehicles
crossing them, it is firstly necessary to define representative loads (called
henceforth APT loads) that are more conservative than other overweight loads
having the same gross weight. This is done in order to ensure that if a bridge is
sufficient for a given APT load, it is automatically adequate for any load less than
or equal to the APT load.
Secondly, it is important to define the possible travel conditions of the vehicle on
the bridge and the potential restrictions. This is necessary because different travel
conditions can place different demands on the bridge.
Therefore, in this Chapter I define the representative loads and the travel
conditions starting from the current conservative procedure adopted by APT DoT
to release the transit permits.
2.2 State of the art
The increase in the number of overweight vehicles travelling on highways has
caused growing interest in the policies for issuing permits (Fiorillo and Ghosn,
2014). Various works on this topic have been developed in recent years and
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included optimization process to obtain optimum paths (Adams et al., 2002;
Osegueda et al., 1999) and procedures for the statistical categorization of
overweight loads (Fiorillo and Ghosn, 2014, Correia et al., 2005; Fu and Hag-
Elsafi, 2000). As mentioned in Section 1.3, three main methods were recently
developed to manage transit permits. Other studies on permit checking of
overweight vehicles were discussed by Fu and Moses (1991), Ghosn (2000),
Bakht and Jaeger (1984), Ghosn and Moses (1987). Phares et al. (2004)
presented a procedure for permit checking of overweight vehicles based on the
bridge load rating using experimental testing.
The first method (Correia et al., 2005) compares the internal forces (bending
moment, shear stress, normal force) caused by design code loads and
overweight vehicles. The methodology requires the knowledge of the transversal
section and longitudinal model of the bridge. Four transversal sections and three
longitudinal models can be chosen for the analyses (Fig. 4). The software BIST
(Bridge Investigation for Special Trucks) developed by Branco and Correia (2004)
calculates, on the basis of the transversal section and longitudinal model chosen,
the transversal distribution load factors and the influence lines necessary for the
comparison of the effects of the design load with those caused by the overweight
vehicle.
Fig. 4: a)Transversal sections and b)longitudinal model considered by software BIST
(Correia and Branco, 2006)
The analyses are performed by placing the loads in each critical section. A
simplification regarding overweight loads is assumed and consists in considering
a) b)
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each group of wheels of the vehicle, which is composed of 7 axles. All the
possible combinations of positions of the loads are tested to determine the
maximum effects in each critical section.
The method calculates the structural safety of the bridge by defining two
simplified safety factors (SSF) evaluated in the critical sections, one for bending
moment and the other for supporting reactions (1).
(1)
where γs (usually equal to γs=1.5) is the safety factor of the code loads, and S(code
loads) and S(overweight loads) are the load effects caused by design loads and overweight
loads respectively. If SSF>1.5, the structural safety of the bridge is greater than
that defined in the design code, and so the crossing of the bridge is safe. Correia
and Branco (2006) states that a lower safety factor for the overweight load may
be adopted, since the exact value of the action is known. As a consequence, the
Portuguese Expressway Authority (Brisa) allows the crossing of the bridge with a
factor of up to γs=1.1, while if γs is between 1.1 and 1.0, the engineer must decide
whether to issue the permit by evaluating the bridge conditions.
The second method, developed by FHWA (1994), estimates the equivalent load
of a generic overweight vehicle. The formula is adopted by the National System of
Interstate and Defense Highways of the U.S.. The generic vehicle is divided into
groups of axles and the equation provides the allowable weight for each group
(Kurt, 2000). The Federal Bridge Formula is the following:
[
] (2)
where W is the overall gross weight on any group of 2 or more consecutive axles,
L is the length of the group, and N is the number of axles in the group.
The allowable gross weight for the overweight vehicle is equal to the sum of the
allowable weights for each axle group (Kurt, 2000). This formula requires many
calculations that depend on the number of axles of the vehicle. In addition, the
equation does not take into account the total gross weight and the total length of
the vehicle, and can therefore be inaccurate especially for long span or multispan
bridges (Vigh and Kollar, 2007). Numerous researchers (Meyerburg et al. (1996),
Nowak et al. (1994), Nowak et al. (1991), Harmon (1982)) have observed that the
overall length of many trucks and the number of axles have significantly increased
since the original bridge formula (2) was developed. This means that the
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allowable weight has significantly increased. For this reason numerous studies to
improve the procedure for issuing transit permits have been done (e.g. Kurt,
2000; Chou, 2005; Ghosn, 2000).
The third method (Vigh and Kollar, 2007) proposes the application of approximate
transversal load distribution to estimate the effects of the loads. The principle of
the method is the following:
The bridge is safe if the internal forces caused by the overweight vehicle are
smaller than those caused by the design load vehicle (Vigh and Kollar, 2007) (3).
(3)
where n is the safety of the bridge, and Edesign load and Eoverweight load are the effects of
the design load and overweight load respectively. If n≥1 the overweight vehicle
can cross the bridge.
The method also takes into account the different location of the load through the
width of the bridge. In fact, Vigh and Kollar (2007) consider the following travel
conditions:
the overweight load is a part of the traffic and crosses close to the exterior
of the bridge;
other traffic loads are not permitted and the overweight load crosses
close to the exterior of the bridge;
the road is closed to free traffic and the overweight load crosses in the
center of the roadway.
The methodology calculates the load that acts in an arbitrary position through the
width of the bridge (Pη), multiplying the load (P) by the distribution factor η
(Goodrich and Puckett, 2000).
(4)
The distribution factors are analogous to the transverse load distribution. Several
methods are available in the literature for the calculation of these coefficients
(Massonnet & Bares, 1968; Guyon, 1946; Courbon, 1950). When the transverse
load distribution coefficients are known, the method suggests that they be used,
otherwise it recommends applying two transverse load distributions which are
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upper and lower approximations of the accurate transverse load distributions (Fig.
5). One of these, which is unknown, is a conservative estimation of the transverse
load distributions. Vigh and Kollar (2007) report the recommended value of these
limits for four different bridge typologies (two girder system, multiple girder
system, solid slab, box). These values were obtained by Janko (1998) by
considering several real bridge superstructures.
Fig. 5: Upper and lower approximation of transverse load distributions (Vigh and Kollar,
2007)
The method points out that since a comparison among loads is performed, the
numerical value of the transverse load distributions are irrelevant and only the
shapes are important.
The main advantage of the permitting procedure based on this last method, is the
minimal input data of the bridge, which are:
the span of the bridge;
the width of the bridge;
type of superstructures of the bridge.
2.3 Current procedure
Nowadays the conservative procedure adopted by the APT’s DoT foresees three
possible transit restrictions based on the total gross weight of the overweight
loads:
1) Free travel up to 44 tons: the vehicle can travel freely with no traffic
restriction – but restrictions on time and number of trips may be applied;
2) Travel with traffic restriction and vehicle speed limit to 30 km/h up
to 60 tons: the road is closed to free traffic, the vehicle is required to
cross the bridge in the center of the roadway and the maximum vehicle
speed is 30 km/h.
3) Travel with traffic restriction and vehicle speed limit to 10 km/h
beyond 60 tons: the road is closed to free traffic, the vehicle is required
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to cross the bridge in the center of the roadway and the maximum
vehicle speed is 10 km/h.
These restrictions are based on empirical information from past crossing of loads
that caused no apparent damage to the bridges. However, these restrictions do
not consider the real capacity of the bridges. In the next sections I analyze
critically this categorization of restrictions and in Chapter 3 I explain a simplified
method that takes into account the capacity of the bridge.
The possible methodologies that can be considered in the analyses are reported
in the following. The differences among the methodologies are in the safety level
that the manager wants to obtain and accept.
1) The overweight load does not cause effects that are more severe than
the original bridge design code loading (the condition satisfies both the
substantial and formal requirements of the design code);
2) the overweight load causes effects that are more severe than the original
bridge design code loading but the exact knowledge of the total gross
weight of the overweight load leads to an estimation of a reliability index
equal to or higher than that foreseen by the design code (the condition
satisfies the substantial but not the formal requirements of the design
code);
3) the manager accepts that the overweight load crosses the bridge with a
reliability index lower than that foreseen by the design code (that is, with
a higher risk), (the condition does not satisfy either the substantial or
formal requirements of the design code).
Table 1 shows the possible methodologies of analysis and travel conditions. The
aim of the present thesis is to analyze only the methodology that satisfies both the
substantial and the formal requirements of the design code (first row of Table 1)
because the APT wants to be protected by legal liabilities involved in possible
catastrophic events. The methodologies that do not satisfy the formal
requirements of the design code are not discussed in the present work (second
and third row of Table 1).
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Table 1: Possible methodologies of analysis and travel conditions
Model
Methodology
Free travel
Bridge crossed in
the center of the
roadway
Limitation of the
speed
As per design
code YES YES YES
With the principle
of the design
code
NO NO NO
Not accepted by
the design load NO NO NO
2.4 Effect of vehicle velocity
In the categorization explained in Section 2.3, the travel restrictions on overweight
loads are based on the vehicle speed. This is justified by assuming that a higher
speed entails a higher dynamic response coefficient. In fact, since the demand on
the bridge caused by a dynamic load is higher than the same load applied
statically, intuitively one can think that a higher speed entails a higher dynamic
response coefficient of the bridge. An examination of the technical literature
(Cantieni 1983, Paultre et al. 1992, Brady et al. 2006, Senthilvasan et al. 2002)
shows that in reality the speed of an overweight vehicle is not directly proportional
to the dynamic coefficient, which depends more on other factors such as
(Cantieni 1983):
the length of the bridge;
the roadway irregularities;
the stiffness of the deck;
the dynamic features of the vehicle.
There is no sense in limiting the speed of the overweight load, if there is no
evidence that by reducing speed, the dynamic coefficient and consequently the
stress on the bridge are also reduced. Moreover, under certain conditions, a lower
speed causes higher stress on the bridge (Fig. 6).
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Fig. 6: Amplification effect as a function of the vehicle velocity (Cantieni, 1983)
2.5 Travel conditions
Based on the observations in Section 2.4, I decided to disregard speed
restrictions, and to assume the following two travel conditions to be used in the
transit permit issuing process:
1) free travel;
2) road closed to free traffic and bridge crossing in the center of the roadway
without speed limits.
As discussed above, the limitation of the speed produces no benefits and so will
not be taken into account in the discussion. Therefore, in accordance with Section
2.3, the possible methodologies of analysis and travel conditions become those
shown in Table 2.
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Table 2: Possible methodologies of analysis and travel conditions considered in the
present thesis
Model
Methodology
Free travel
Bridge crossed in
the center of the
roadway
Limitation of the
speed
As per design
code YES YES NO
With the principle
of the design
code
NO NO NO
Not accepted by
the design load NO NO NO
2.6 APT overweight vehicle models
In general, permits for overweight vehicles are issued by transportation agencies
based on the total gross weight, the maximum axle load, the distribution of axle
weights and the axle spacing (see for example Federal Highway Administration
(2012)). One of the main problems is the high variability of axle loads and
spacing. A set of predefined overweight load configurations, the APT loads, is
therefore defined; these configurations are more conservative than other axle
combinations having the same gross weight. If a bridge is sufficient for a given
APT load, it is automatically adequate for any load less than or equal to the APT
load. Currently, the Italian highway code (D.L. 30/04/1992) places the following
limits on the free movement of vehicles: maximum total weight 44 tons (440 kN);
maximum axle load 13 tons (130 kN); minimum axle spacing 1.3 m. The APT
requires that these axle weight and spacing limits be respected for extra-legal
vehicles, whatever their gross weight. Therefore, the APT load model reproduces
a multi-axle load in the most unfavorable configuration considering different gross
weights: a set of 130kN concentrated loads spaced at 1.3 m, applied on a 3.0 m
wide lane. Similarly to the provisions of the 1990 Italian design code for bridges
(D.M. 04/05/1990), the overweight vehicle model is applied in conjunction with a
uniformly distributed load of 30kN m-1
, 6 m ahead of the first axle and 6 m behind
the last axle. In addition the APT DoT required that two additional models be
taken into account: 6 axles by 12 tons = 72 tons (720kN) and 56 tons (560kN) due
to specific real overweight loads. All APT loads are shown in Table 3. The
concentrated forces represent the axles of the overweight load, and the uniformly
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distributed load of 3 t/m represents the traffic in front of and behind the overweight
load.
Table 3: APT loads
APT Load APT Load pattern
56 tons
5 axles by 13
tons
65 tons
6 axles by 12
tons
72 tons
6 axles by 13
tons
78 tons
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2.7 Conclusions
The objective of the APT DoT is to provide simple and practical rules for deciding
whether an overweight load permit can be issued or not, and if so, under what
restrictions. The current procedure of the APT foresees conservative limits and
takes into account the vehicle velocity. Examining the technical literature it is clear
that there is no sense in limiting the speed of the overweight load as there is no
evidence that by reducing speed, the stress on the bridge is reduced. From these
considerations only two travel conditions are possible: free travel and crossing the
bridge in the center of carriageway.
Representative loads (APT loads), which are more conservative than other
overweight loads having the same gross weight, are defined. If a bridge is
sufficient for a given APT load it is automatically adequate for any load less than
or equal to the APT load. Following this approach it is not necessary to verify a
7 axles by 13
tons
91 tons
8 axles by 13
tons
104 tons
9 axles by 13
tons
117 tons
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bridge every time a different load distribution is acting. The definition of the
representative loads has been reached by analyzing the current Italian code and
considering the minimum axle spacing, and the maximum axle load of the
vehicles foreseen by the code.
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3 CRITERIA FOR RE-ASSESSING BRIDGES FOR
OVERWEIGHT LOADS
3.1 Introduction
In principle, assessing a bridge to any of the APT load models and travel
conditions would require full formal re-evaluation of its capacity, carried out by a
professional engineer. In practice, direct re-assessment of the 1000 bridges in the
territory is a very expensive task, and it is in many cases unnecessary. To guide
the cost issue, the approach followed is to estimate the capacity of the stock
using a simplified and conservative approach first, and then refining the analysis
only if a higher load rating is required.
In this Chapter I show the procedure to evaluate whether a bridge can withstand
the APT loads defined in Chapter 2. In particular, I illustrate the multi-level
assessment protocol conceived to estimate the response of the bridge stock to
overweight loads.
3.2 Assessment principle
The assessment procedure includes four simplified levels of refinement, from
Level 0 to Level 3, as summarized in Table 4. All methods are based on the
following principle:
The bridge is rated for an overweight load if it is demonstrated, even
conservatively, that the overweight load does not cause effects that are more
severe than the original bridge design code loading.
30
In practice the procedure aims to demonstrate that the new overweight load
condition is not worse than the most critical load condition associated with the
original design assumption or as-built situation, and therefore the overweight load
does not reduce the original design safety level. It is believed that this principle
protects the APT against legal liability in the case of a failure, as following this
principle the APT authorizes travel only of those vehicles whose effects are less
than or equal to those expected under the design assumptions.
Table 4: Level of refinement of bridge assessment under overweight loads
Assessment Level Capacity Models Calculation Models
Level 0 Bridge is assumed
verified with no overstrength
Statically determinate condition is assumed
Level 1 As per design
As per design Level 2
Refined model; load
redistribution is allowed,
provided that the ductility
requirements are fulfilled.
Level 3
Material properties can
be updated based on in-
situ testing and
observations
The fundamental hypothesis of the assessments is reported in the following:
the contemporary presence of more than one overweight load on the
bridge is not allowed.
In the case of free travel we evaluate that the condition above is still valid,
because the contemporary presence of more than one overweight load is unlikely.
Moreover, if this situation of more than one vehicle is considered, the result of the
method would be too conservative. The hypothesis is automatically satisfied in
the case of bridge crossed in the center of the roadway.
3.3 Levels of assessment
As summarized in Table 4, three levels of refinement of bridge assessment are
foreseen to verify the bridge under overweight loads.
Level zero assessment is a conservative estimate of the bridge capacity based on
its geometry and the design code it was originally designed to. The fundamental
hypothesis of Level 0 assessment is reported in the following:
31
the bridge was built exactly to the nominal design load (in other words,
with no overstrength).
The bridge is automatically rated for the overweight load if it can be demonstrated
that the stresses it creates are everywhere lower than or equal to those
considered at the design stage.
The APT approach:
1) replaces the design traffic lane with the APT load;
2) evaluates the difference in demand between the new and old loadings.
If the stress under the APT load is lower than the design load stress, it is logical to
infer that the bridge is able to withstand the APT load. For girder bridges, shear
and bending moment in the deck are assessed.
The basic assumption in Level 0 assessment is that the bridge was built with no
shear or moment overstrength at any section of the deck: this assumption is
extremely conservative and normally not realistic. Bridge members are normally
slightly oversized and very often the designer assumes conservative capacity
models. In addition the mechanical properties of the materials in the as-built
situation could be better than those specified by the designer. When a bridge is
not sufficient according to Level 0 assessment, the assessment proceeds to a
higher level of refinement and is carried out by a professional engineer based on
analysis of the available design documentation.
At Level 1 the evaluator is required to examine the design documents to verify
whether the critical structural members are oversized. This level of assessment is
based only on analysis of the existing documents, without revising assumed
material properties or the calculation model.
At Level 2 the evaluator reassesses the bridge capacity using more refined
models than those used by the original designer. Often, a capacity increase is
achieved by considering inelastic behavior and spatial stress redistribution.
At Level 3 the evaluator can update the characteristic values of the variables used
in the assessment, based on the results of material testing and observations. The
procedure leaves the evaluator free to test the materials, without specifying the
minimum number of samples or the type of test. However, if the evaluator wants
to use the test results quantitatively, a Bayesian probabilistic update technique
should be applied. This can be done even if the design documentation is not
available: in this case, an accurate survey of bridge geometry and extensive
material sampling are needed.
32
3.4 Procedures for assessment
Although the definition of levels of assessment is general, the procedure for
assessment is different in the case of girder bridges and arch bridges. The reason
lies in the different resistance mechanisms of the two typologies. For arch
bridges, in contrast with girder bridges, the capacity is independent of the material
strength (for example the compressive strength of the masonry) but depends on
the equilibrium of the forces acting upon it. In fact, if the form of the arch bridge is
equal to the funicular form generated by the external forces, its capacity tends to
infinity. Therefore, in Section 3.4.1 and 3.4.2 the procedure for assessment of
girder and arch bridges is shown separately. In these sections only the procedure
of Level 0 assessment is reported. The other levels depend on the specific
verifications of the bridge and these are shown in the case studies of Chapter 4.
In the assessment levels we assume that the widths of the design load and the
APT load are identical (as suggested also by Vigh and Kollar, 2006). The Italian
codes (see for example Circolare 1962 or Circolare 1952) foresee two typology of
design loads:
civil loads (width=3.0m);
military loads (width=3.5m).
Therefore, in our case, the width of the APT load can be 3.0m or 3.5m.
3.4.1 Girder bridges
The procedure for assessment involves an estimation of the capacity of the
bridge, using Level 0 assessment first, and then refining the assessment level if a
higher assessed capacity is required. Level 0 analysis is simple for the free travel
condition. In this case the design situation is compared to a load condition where
one traffic lane is replaced with the APT load. As discussed above, the method is
based on the following principle:
The bridge is rated for an overweight load if it is demonstrated, even
conservatively, that the overweight load does not cause effects that are more
severe than the original bridge design code loading.
33
Since the method compares effects, the following terms can be neglected in the
calculations:
dead load of the bridge;
live loads outside the lane replaced.
The reason is that these loads are the same for both the overweight and the APT
loadings, and so their effects are irrelevant to the evaluation and can be
disregarded. In fact, if S0 is the demand caused by the overweight load, which can
represent for example the bending moment or shear stress, SAPT is that caused by
the APT load, and G is that caused by the dead load, the increase in the demand
ΔS is independent of the dead load, and is equal to (5):
(5)
Similarly, the effects of live loads outside the lane replaced are identical in the two
situations, so these live loads can also be disregarded too. In fact, only one traffic
lane is replaced by the APT load and consequently the demand caused by any
live loads is the same in the two situations (6).
(6)
where S1 and S2 are the demands caused by the second and the third traffic lane
respectively.
In contrast, partial factors (γ) or dynamic coefficients (Φ) considered at the design
stage influence the increase in the demand (7).
[ Φ]
[ Φ] Φ
(7)
However, these values do not change the sign of the result but only its value. This
means that the result of the verification (positive or negative) is identical even if
partial factors or dynamic coefficients are not taken into account. For this reason
the comparison is conducted using the nominal load value, disregarding any
34
another coefficient. If we need to quantify the extent of bridge understrength with
respect to the overweight load as absolute value, it is also necessary to consider
the partial factors or the dynamic coefficients. In contrast, if this deficiency is
evaluated as a relative value (ΔS/S0), as in Section 3.5, these coefficients can also
be neglected (8).
[ Φ]
Φ
[ Φ]
Φ
Φ
Φ
(8)
Moreover, when the shape of the design load and APT load are similar, the static
scheme of the bridge can also be neglected in the analyses.
Let us consider a simply supported beam and a fixed beam as shown in Fig. 7
and in Fig. 8.
Fig. 7: Distributed load on a)simply supported beam and b)fixed beam
Fig. 8: Concentrated load on a)simply supported beam and b)fixed beam
The diagram of the bending moment of the fixed beam can be seen as a shift of
that obtained for the simply supported beam. This is valid not only for distributed
loads but for each type of load (e.g. concentrated loads) (Fig. 9 and Fig. 10).
This means that when the original design load produces static stresses higher
than the new overweight vehicle in a simply supported span, the design stresses
will be higher with a statically indeterminate boundary condition. Since the method
Q
LL
Q F
LL
F
Q
LL
Q F
LL
F
35
foresees a comparison between the effects caused by design and overweight
loads, the static scheme, whether continuous or simply supported, is irrelevant, so
a simply supported condition can always be assumed.
Fig. 9: Bending moment diagram caused by a distributed load in the case of a)simply
supported beam, b)fixed beam, c)comparison.
Fig. 10: Bending moment diagram caused by a concentrated load in the case of a)simply
supported beam, b)fixed beam, c)comparison.
In conclusion, in the case of free travel of an overweight load, the assessment is
made by comparing the effects of the original design and APT loads on a single
lane simply supported beam.
a)
c)
b)
a)
c)
b)
36
The parameters that control the assessment are:
the span length L,
the design code adopted by the designer,
which in turn is related to the date of construction of the bridge.
In practice, Level 0 assessment foresees the calculation, for each section of the
beam, of the maximum bending moment and shear caused by the design load.
Subsequently, the design year of the bridge must be known, to obtain the design
loads provided by the design code. Then the same procedure is applied to the
APT load. If in all sections of the deck the bending moment and the shear caused
by APT load are equal to or lower than those caused by the design load, the
verification is positive (Fig. 11).
Fig. 11: Example of Level 0 assessment
In the case of restricted travel, the effects from the original traffic lanes must be
compared to the single lane APT load applied in the center of the carriageway. In
this case, to compare the effects, we must take account of the transverse load
redistribution between the various girders using, for example, the Massonnet-
Guyon-Bares theory (Massonnet C., & Bares R., 1968).
37
Fig. 12 illustrates the procedure for determining whether the effect of the
overweight load centered on the carriageway is more or less critical than any
combination of design traffic lanes. Fig. 1a shows the cross section of the bridge
deck loaded with a two-lane design condition, with the Massonnet-Guyon-Bares
transverse distribution coefficients multiplied by the corresponding load
highlighted. The effect of the second design load configuration is illustrated in Fig.
12b, while Fig. 12c shows the envelope of design conditions. The maximum
allowable APT load, applied in the center of the carriageway, is that which has
effects less than or equal to the envelope of the design load effects (Fig. 12d).
Therefore, knowing the transverse distribution coefficients of the design load, the
maximum allowable APT load in the center of the carriageway can be determined
for any deck width or any bridge.
a) b)
c) d)
Fig. 12: a) Massonnet-Guyon-Bares coefficients multiplied by the relative load, first load
design condition b) second load design condition, c) envelope of design condition, d)
comparison with the effect of an APT overweight load.
38
In summary, in the case of restricted traffic, Level 0 assessment requires
knowledge of:
the transverse distribution coefficients of the design load;
the span length;
the design code.
As an alternative to the Massonnet-Guyon-Bares method, the Courbon method
can be used to calculate the transverse distribution coefficients of the design load.
The Courbon method is a specific case of the Massonnet-Guyon-Bares method
(Petrangeli, 1996) in which the hypotheses are the following:
the torsional stiffness of the deck is equal to zero;
the longitudinal stiffness of the deck tends to infinity.
The method is not always in favor of safety, and therefore the final analyses
should be performed using Massonnet-Guyon-Bares method. However, the
Courbon method is a valid tool to estimate the lack of capacity of bridges, and
with this aim the results shown in Section 6.4 were obtained using this method.
When evaluating a bridge deck made with a concrete slab, the Courbon
coefficient is equal to (9):
(
) (9)
where r(x) is the transverse distribution coefficient as a function of the transverse
coordinate x (x=L/2 in the center of the carriageway), e is the load eccentricity
from the center of the carriageway, and L is the width of the deck.
Fig. 13 shows the Courbon transverse distribution of load as a function of one
moving concentrated force along the transverse section of the deck slab.
It can be noted from equation (9) and Fig. 13 that r(x) is equal to one in the center
of the carriageway (x=L/2) for each value of eccentricity e. This means (Fig. 14)
that the minimum value of the envelope caused by a number of loads is in the
center of the carriageway. Moreover it can be noticed that the minimum value of
the envelope is equal to the sum of the loads. In the case of a bridge crossed in
the center of carriageway, the allowable APT load is equal to the sum of the
design loads (Fig. 14).
39
Fig. 13: Courbon transverse distribution of load as a function of one moving concentrated
force
Fig. 14: Maximum allowable APT load in the case of bridge crossed in the center of
carriageway
(10)
Equation (10) is valid only in the case of absence of the traffic divider. When a
traffic divider is present the bridge is crossed in the center of the semi-
carriageway and it is necessary to use the formulation explained in Section 3.4
replacing the Massonnet-Guyon-Bares coefficients with the Courbon coefficients
(Fig. 15).
4P
P
1P
2P
L L
0.5P
1P
P
2.5P
L/4x
L/2
P
1P
L
3/4L
2.5P
P
1P
0.5P
L L
2P
1P
P
4P
40
Fig. 15: Maximum allowable APT load in the case of bridge crossed in the center of
carriageway with traffic divider
3.4.2 Arch bridges
For arch bridges, the approach shown in Section 3.4.1 has to be modified as the
resistance mechanisms of the two types of bridges is different. For girder bridges,
the procedure involves comparing, for each section of the beam, the maximum
bending moment and shear caused by the design load and by the APT load. This
procedure cannot be implemented for arch bridges because their capacity
depends only on the equilibrium of the forces acting upon them (Heyman, 1982).
The principle defined in Section 3.2 is still valid but the approach must be
changed. In fact, the capacity of the arch element is independent of the material
strength but depends on the equilibrium of the forces acting upon it. If the form of
the arch bridge is equal to the funicular form generated by the external forces, its
capacity tends to infinity. This means that the capacity depends on the form of the
load (axle spacing and axle loads) and not on the total gross weight (Fig. 16). Fig.
16 shows thrust lines generated by a uniform load q and q+q by supposing the
form of the arch equal to the funicular form (in this case a parabolic shape). It can
be noticed that in both cases the thrust line is identical.
The verification of the arch element can be performed by checking that the thrust
line is always within the middle third. This ensures that the cross-sections of the
arch are always compressed. Heyman, (1982) stated that to verify the arch
element it is sufficient demonstrate that there exist a satisfactory line of thrust that
balances the given loading. In other words it is sufficient to find a thrust line in
equilibrium with the external loading which lies everywhere within the arch ring
(Heyman, 1982). This principle is used in Level 2 assessment, where the
equilibrium analysis of the arch is performed.
41
a) b)
Fig. 16: a)thrust line generated by a uniform load q, b)thrust line generated by a uniform
load q+q
For Level 0 assessment it is sufficient to compare the eccentricity of the thrust line
caused by the design load with that caused by the APT load. If in each section of
the arch, the eccentricity caused by the design load is greater than that caused by
the APT load, the bridge is safe. In fact, the eccentricity caused by the design
load represents the minimum thickness of the arch. An example of Level 0
assessment for arch bridges is reported in the following.
In the voussoir arch of Fig. 17, the connections between the arch rib and the
abutments can be considered made by frictionless pins as reported in the
literature (for example Heyman, 1982).
Fig. 17: Side view of voussoir arch bridge
If the loading F is located in the correspondence of the crown (Fig. 18), the total
bending moment M can be determined by performing the difference between the
Thrust line
q
Thrust line
42
bending moment caused by external forces (R·x) and by the abutment thrust (H·y)
(11).
Fig. 18: Schematization of the arch bridge
(11)
This means that the total bending moment (Fig. 19c) can be determined
graphically by minimizing the difference between the bending moment caused by
external forces (Fig. 19a) and a moment with the same shape of the arch (Fig.
19b). In fact, the shape of the arch represents the bending moment caused by the
abutment thrust.
Fig. 19: Bending moment in the arch rib, a)caused by one concentrated force, b)caused by
the horizontal component of the abutment thrust c)total bending moment.
a)
b)
c)
43
Similarly to girder bridges, Level 0 assessment foresees the calculation, for each
section of the arch rib, the envelope of the bending moment (Fig. 21c) caused by
the design load (considering the worst position of the load (see for example Fig.
20)).
Fig. 20: Schematization of the arch bridge with a moving load
Fig. 21: Bending moment in the arch rib, a)caused by one moving concentrated force,
b)caused by the horizontal component of the abutment thrust c)envelope of the total
bending moment.
a)
b)
c)
44
The same procedure is applied to the APT load. If in all sections of the deck the
bending moment caused by APT load is equal to or lower than that caused by the
design load, the verification is positive.
The approach discussed above requires the knowledge of the shape of the arch.
In the case of the APT bridge stock these data are not known and then an
approximation of the arch shape must be done to apply Level 0 assessment. In
this assessment a parabolic shape of the arch was chosen.
For Level 2 assessment this is not necessary because the exact shape can be
determined performing a geometric survey of the arch.
It is important to point out that for arch bridges, Level 1 assessment makes no
sense because by maintaining the same calculation model as per design the
result obtained would involve an increase either in the abutment thrust or in the
eccentricity and therefore the same results as the Level 0 assessment would be
obtained. Moreover, applying Level 3 assessment it is not possible to obtain a
decrease in the lack of capacity shown after Level 0 or Level 2 assessments. The
reason is that the mechanical properties of materials are not involved in the
resistance mechanisms of arch bridges. In fact, the capacity is independent of the
material strength but depends on the equilibrium of the forces acting upon it.
3.5 Lack of capacity of substandard bridges
To classify substandard bridges, we need to quantify the extent of their
deficiency. A useful index here is the critical live load multiplier η, defined as the
ratio between the stresses caused by the APT load and those due to the design
load (12).
(12)
where S0 is the effect of the live load and SAPT is the effect of the APT load.
For girder bridges there will be a critical load multiplier for bending moment and
shear stress, and these indices are calculated for each section of the span. The
coefficient depends on the location of the section where the ratio is calculated.
Therefore, the number of critical live load multipliers is equal to the number of
sections where the comparison is performed. However, the most significant
values are the maximum critical live load multiplier (maxη) and the critical live load
multiplier calculated where the stress is maximum (ηmax). The latter is defined as
the ratio between the maximum stress caused by the APT load and that due to
the design load, and is independent of the location where the maximum stresses
45
are. ηmax gives an in depth understanding of the deficiency of the bridge even if
the location of the sections where the comparison is done is different. In fact, in
most cases, these sections are close to each other and the capacity is piecewise
constant. This means that even if the locations of the maximum stresses are not
the same, we can assume this to be the case in order to estimate the response of
the bridge. We can therefore calculate the critical live load multipliers for bending
moment and shear stress. For each bridge the following indexes were calculated:
maximum critical live load multiplier for shear stress (maxηt);
maximum critical live load multiplier for bending moment (maxηm);
critical live load multiplier for maximum shear stress (ηt,max);
critical live load multiplier for maximum bending moment (ηm,max).
As discussed in Section 3.4.1, the parameters that control the assessment are
the span length and the date of construction of the bridge. Thus, the safety
factors, the dynamic coefficient and the static scheme of the bridge are irrelevant
in the comparison. This is to say that it is possible to express the critical live load
multipliers as a function of these parameters. In fact, Fig. 22 illustrates the critical
live load multipliers as a function of the design code adopted by the designer, the
span length, and the APT load (in the case of free travel). In Fig. 22 only two
design codes are included (D.M. 1990 and Circolare 1962).
46
a)
Max η ηmax Max η ηmax Max η ηmax Max η ηmax Max η ηmax Max η ηmax
V 0.533 0.533 0.650 0.650 0.650 0.650 0.650 0.650 0.650 0.650 0.650 0.650M 0.533 0.533 0.650 0.650 0.650 0.650 0.650 0.650 0.650 0.650 0.650 0.650V 0.604 0.576 0.737 0.702 0.737 0.702 0.737 0.702 0.737 0.702 0.737 0.702M 0.604 0.533 0.737 0.650 0.737 0.650 0.737 0.650 0.737 0.650 0.737 0.650V 0.604 0.586 0.737 0.737 0.737 0.737 0.737 0.737 0.737 0.737 0.737 0.737M 0.654 0.582 0.823 0.732 0.823 0.732 0.823 0.732 0.823 0.732 0.823 0.732V 0.586 0.558 0.737 0.711 0.737 0.711 0.737 0.711 0.737 0.711 0.737 0.711M 0.595 0.585 0.749 0.737 0.749 0.737 0.749 0.737 0.749 0.737 0.749 0.737V 0.586 0.546 0.755 0.755 0.755 0.755 0.755 0.755 0.755 0.755 0.755 0.755M 0.572 0.563 0.803 0.718 0.803 0.718 0.803 0.718 0.803 0.718 0.803 0.718V 0.586 0.540 0.819 0.819 0.819 0.819 0.819 0.819 0.819 0.819 0.819 0.819M 0.558 0.551 0.832 0.780 0.832 0.780 0.832 0.780 0.832 0.780 0.832 0.780V 0.568 0.535 0.867 0.867 0.886 0.886 0.886 0.886 0.886 0.886 0.886 0.886M 0.549 0.544 0.883 0.841 0.883 0.841 0.883 0.841 0.883 0.841 0.883 0.841V 0.558 0.533 0.900 0.900 0.950 0.950 0.957 0.957 0.957 0.957 0.957 0.957M 0.543 0.540 0.917 0.881 0.945 0.900 0.948 0.900 0.948 0.900 0.948 0.900V 0.551 0.550 0.924 0.924 0.997 0.997 1.031 1.031 1.031 1.031 1.031 1.031M 0.549 0.536 0.944 0.910 0.993 0.956 1.025 0.984 1.025 0.984 1.025 0.984V 0.573 0.573 0.940 0.940 1.029 1.029 1.085 1.085 1.108 1.108 1.108 1.108M 0.570 0.534 0.961 0.932 1.026 0.998 1.083 1.051 1.101 1.051 1.101 1.051V 0.594 0.594 0.947 0.947 1.049 1.049 1.121 1.121 1.164 1.164 1.178 1.178M 0.592 0.532 0.974 0.949 1.055 1.031 1.133 1.103 1.164 1.124 1.171 1.132V 0.607 0.607 0.951 0.951 1.060 1.060 1.145 1.145 1.203 1.203 1.236 1.236M 0.606 0.530 0.986 0.962 1.080 1.058 1.172 1.144 1.222 1.185 1.232 1.213V 0.620 0.620 0.953 0.953 1.065 1.065 1.159 1.159 1.230 1.230 1.277 1.277M 0.618 0.534 0.994 0.973 1.100 1.079 1.200 1.178 1.268 1.234 1.298 1.280V 0.633 0.633 0.956 0.956 1.068 1.068 1.166 1.166 1.247 1.247 1.306 1.306M 0.632 0.546 0.996 0.982 1.115 1.097 1.219 1.206 1.301 1.275 1.351 1.336V 0.642 0.642 0.958 0.958 1.071 1.071 1.171 1.171 1.257 1.257 1.326 1.326M 0.641 0.566 0.997 0.990 1.123 1.113 1.231 1.230 1.324 1.310 1.389 1.383V 0.647 0.647 0.959 0.959 1.073 1.073 1.174 1.174 1.262 1.262 1.338 1.338M 0.646 0.582 0.997 0.995 1.127 1.124 1.248 1.248 1.341 1.338 1.421 1.421V 0.649 0.649 0.961 0.961 1.074 1.074 1.176 1.176 1.266 1.266 1.345 1.345M 0.649 0.594 0.997 0.995 1.132 1.129 1.259 1.259 1.360 1.357 1.448 1.448V 0.650 0.650 0.963 0.963 1.075 1.075 1.177 1.177 1.269 1.269 1.349 1.349M 0.654 0.603 0.997 0.994 1.134 1.131 1.265 1.265 1.372 1.369 1.468 1.468V 0.650 0.649 0.964 0.964 1.076 1.076 1.178 1.178 1.270 1.270 1.353 1.353M 0.660 0.614 0.997 0.993 1.133 1.130 1.266 1.266 1.378 1.375 1.480 1.480V 0.650 0.648 0.965 0.965 1.076 1.076 1.178 1.178 1.271 1.271 1.355 1.355M 0.663 0.623 0.997 0.992 1.132 1.128 1.264 1.264 1.380 1.378 1.487 1.487V 0.650 0.648 0.967 0.967 1.077 1.077 1.178 1.178 1.271 1.271 1.355 1.355M 0.664 0.631 0.997 0.991 1.130 1.126 1.261 1.261 1.379 1.376 1.489 1.489V 0.650 0.649 0.968 0.968 1.077 1.077 1.178 1.178 1.271 1.271 1.356 1.356M 0.664 0.638 0.997 0.990 1.128 1.124 1.258 1.258 1.376 1.373 1.488 1.488V 0.651 0.651 0.969 0.969 1.077 1.077 1.178 1.177 1.271 1.270 1.356 1.355M 0.662 0.645 0.997 0.990 1.126 1.122 1.254 1.254 1.372 1.369 1.484 1.484V 0.653 0.653 0.970 0.970 1.077 1.076 1.178 1.176 1.271 1.268 1.356 1.354M 0.661 0.651 0.997 0.989 1.124 1.120 1.251 1.251 1.368 1.365 1.479 1.479V 0.655 0.655 0.971 0.971 1.077 1.076 1.178 1.175 1.271 1.267 1.356 1.352M 0.661 0.657 0.997 0.989 1.122 1.118 1.247 1.247 1.363 1.361 1.474 1.474
DM_04_05_1990
21
22
23
24
25
15
16
17
18
19
20
9
10
11
12
13
14
3
4
5
6
7
8
1
2
9x13 t56 t 5x13 t 6x13 t 7x13 t 8x13 tSPAN
(m)
47
b)
Fig. 22: Critical live load multipliers as a function of the design code [a)D.M. 1990 and
b)Circolare 1962] adopted by the designer, the span length, and the APT load.
Max η ηmax Max η ηmax Max η ηmax Max η ηmax Max η ηmax Max η ηmax
V 0.561 0.561 0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684M 0.561 0.561 0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684V 0.561 0.552 0.684 0.672 0.684 0.672 0.684 0.672 0.684 0.672 0.684 0.672M 0.561 0.561 0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684V 0.598 0.598 0.752 0.752 0.752 0.752 0.752 0.752 0.752 0.752 0.752 0.752M 0.586 0.545 0.738 0.686 0.738 0.686 0.738 0.686 0.738 0.686 0.738 0.686V 0.674 0.674 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858 0.858M 0.673 0.656 0.847 0.825 0.847 0.825 0.847 0.825 0.847 0.825 0.847 0.825V 0.716 0.716 0.990 0.990 0.990 0.990 0.990 0.990 0.990 0.990 0.990 0.990M 0.721 0.710 0.980 0.906 0.980 0.906 0.980 0.906 0.980 0.906 0.980 0.906V 0.725 0.696 1.055 1.055 1.055 1.055 1.055 1.055 1.055 1.055 1.055 1.055M 0.750 0.742 1.061 1.051 1.061 1.051 1.061 1.051 1.061 1.051 1.061 1.051V 0.725 0.668 1.081 1.081 1.106 1.106 1.106 1.106 1.106 1.106 1.106 1.106M 0.769 0.763 1.187 1.179 1.187 1.179 1.187 1.179 1.187 1.179 1.187 1.179V 0.725 0.650 1.099 1.099 1.160 1.160 1.168 1.168 1.168 1.168 1.168 1.168M 0.782 0.778 1.277 1.270 1.298 1.297 1.298 1.297 1.298 1.297 1.298 1.297V 0.725 0.658 1.108 1.105 1.191 1.191 1.232 1.232 1.232 1.232 1.232 1.232M 0.762 0.743 1.292 1.261 1.345 1.325 1.398 1.364 1.398 1.364 1.398 1.364V 0.725 0.668 1.108 1.097 1.200 1.200 1.266 1.266 1.292 1.292 1.292 1.292M 0.731 0.717 1.275 1.252 1.356 1.341 1.438 1.412 1.438 1.412 1.438 1.412V 0.715 0.675 1.108 1.077 1.201 1.193 1.275 1.275 1.323 1.323 1.339 1.339M 0.708 0.698 1.262 1.244 1.365 1.352 1.467 1.446 1.486 1.475 1.506 1.484V 0.696 0.679 1.108 1.063 1.201 1.185 1.280 1.280 1.345 1.345 1.382 1.382M 0.691 0.683 1.253 1.238 1.371 1.361 1.490 1.472 1.535 1.525 1.580 1.562V 0.687 0.687 1.108 1.056 1.201 1.180 1.284 1.284 1.363 1.363 1.415 1.415M 0.695 0.677 1.245 1.233 1.376 1.368 1.507 1.493 1.573 1.564 1.638 1.623V 0.699 0.699 1.108 1.055 1.201 1.180 1.288 1.288 1.377 1.377 1.442 1.442M 0.698 0.681 1.239 1.225 1.380 1.369 1.521 1.505 1.603 1.591 1.685 1.667V 0.704 0.704 1.107 1.051 1.201 1.175 1.288 1.284 1.379 1.379 1.454 1.454M 0.703 0.692 1.234 1.210 1.384 1.360 1.533 1.503 1.629 1.602 1.724 1.691V 0.708 0.708 1.100 1.050 1.201 1.174 1.288 1.285 1.382 1.382 1.464 1.464M 0.708 0.699 1.220 1.195 1.375 1.350 1.530 1.500 1.636 1.607 1.742 1.707V 0.711 0.711 1.093 1.053 1.198 1.177 1.289 1.289 1.387 1.387 1.474 1.474M 0.715 0.701 1.206 1.175 1.365 1.333 1.525 1.487 1.640 1.602 1.754 1.710V 0.714 0.714 1.083 1.059 1.190 1.183 1.295 1.295 1.395 1.395 1.484 1.484M 0.721 0.703 1.195 1.159 1.358 1.319 1.521 1.475 1.643 1.596 1.765 1.712V 0.717 0.717 1.067 1.066 1.190 1.190 1.303 1.303 1.405 1.405 1.496 1.496M 0.724 0.711 1.180 1.149 1.342 1.308 1.505 1.465 1.632 1.592 1.759 1.713V 0.722 0.722 1.075 1.075 1.199 1.199 1.313 1.313 1.416 1.416 1.509 1.509M 0.726 0.718 1.170 1.143 1.331 1.300 1.492 1.457 1.623 1.588 1.755 1.714V 0.728 0.728 1.086 1.086 1.209 1.209 1.323 1.323 1.428 1.428 1.523 1.523M 0.734 0.726 1.165 1.141 1.323 1.296 1.482 1.452 1.616 1.584 1.751 1.714V 0.737 0.737 1.098 1.098 1.221 1.221 1.335 1.335 1.441 1.441 1.537 1.537M 0.745 0.736 1.163 1.141 1.320 1.296 1.477 1.450 1.612 1.583 1.747 1.715V 0.746 0.746 1.110 1.110 1.233 1.233 1.348 1.348 1.455 1.455 1.552 1.552M 0.753 0.745 1.158 1.144 1.313 1.297 1.468 1.449 1.602 1.582 1.736 1.715V 0.756 0.756 1.123 1.123 1.247 1.247 1.362 1.362 1.469 1.469 1.568 1.568M 0.760 0.753 1.156 1.144 1.309 1.295 1.462 1.447 1.596 1.579 1.729 1.711V 0.768 0.768 1.137 1.137 1.260 1.260 1.376 1.376 1.484 1.484 1.584 1.584M 0.768 0.762 1.157 1.147 1.308 1.297 1.459 1.446 1.592 1.578 1.725 1.710
21
22
23
24
25
15
16
17
18
19
20
9
10
11
12
13
14
3
4
5
6
7
8
1
2
9x13 t56 t 5x13 t 6x13 t 7x13 t 8x13 t
CIRC_1962SPAN
(m)
48
In Table 5 the association of the cell colors used in Fig. 22 with critical live load
multiplier (η) is shown.
Table 5: Critical live load multiplier and assigned colors
Critical live load multiplier Color
η ≤1
1 < η ≤ 1.1
1.1 < η ≤ 1.3
1.3 < η ≤ 1.6
η > 1.6
Knowing the critical live load multipliers for each bridge, it is possible to define a
priority ranking for future assessments. If η is slightly greater than 1, there is a
high probability that by applying Level 1 assessment the verifications become
positive. As η increases, this probability is smaller and smaller.
It is worth emphasizing that a critical live load multiplier η of less than 1.0 can
have different impacts on the overall safety of the bridge, depending on the
magnitude of the dead load. For example, a live load multiplier of 0.90 could be
critical for short span steel girder bridges, where the live load is dominant, but is
likely to be much less critical for a long span reinforced concrete bridge, where
the dead load dominates.
An index that better reflects bridge understrength with respect to the overweight
load is the lack of capacity factor α, defined as the percentage of additional
capacity R needed to safely carry the overweight load. For girder bridges this
coefficient indicates how much the most critical bending moment or shear stress
must increase for positive verification of the bridge. There is a direct relationship
between indices α and η. If R is the capacity of the bridge at a specific limit state
and SAPT is the overweight load demand, the lack in capacity α of the bridge can
be expressed as follows (13):
0 0
0 0
( 1) 1
1 1
APT G S G SS R S
R R G S G S
(13)
where G is the effect of the dead load and S0 is the effect of the live load. Equation
(13) shows that computing index requires knowledge of the G/S0 ratio for the
bridge, and this in turn depends on the bridge characteristics as included in the
design documentation, normally not known for a Level 0 assessment. To estimate
index at Level 0, a practical solution is to provide approximate expressions for
49
the dead loads of various bridge structures and span lengths. To do this, it is
necessary to evaluate, for each main deck type, the weight of the bridge in a
number of case studies (Table 6) and then to extrapolate the relative interpolating
curve (Fig. 23a). This curve provides the weight of a lane width of 3 meters as a
function of the bridge span.
The graphs in Fig. 23 show the dead load of a 3 m wide lane calculated for a
sample of bridges of differing construction technologies, with proposed fitting
curves that can be used to estimate the dead load once the bridge span is known.
Table 6: Weight of reinforced concrete bridges
Reinforced concrete girder bridge
Span length
[m]
P/L (lane of 3m)
[kN/m]
3.0 34
7.4 36
12.4 37
20.0 37
25.0 43
33.7 67
a) reinforced concrete girder b) precast reinforced concrete girder
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50
de
ad lo
ad g
[kN
∙m‾¹
]
span length L [m]
0
10
20
30
40
50
60
70
80
0 10 20 30 40
dea
d lo
ad g
[kN
∙m‾¹
]
span length L [m]
50
c) steel girder d) reinforced concrete slab
e) precast reinforced concrete slab f) reinforced concrete box girder
g) steel-concrete box girder f) arch bridges
Fig. 23: Bridge span to dead load relationship for various bridge technologies: experimental
samples and fitting curves
3.6 Risk interpretation of parameter
This section provides the risk interpretation of parameter α in terms of the
probability of failure of a substandard bridge caused by the APT load.
0
10
20
30
40
50
60
70
80
0 20 40 60 80
de
ad lo
ad g
[kN
∙m‾¹
]
span length L [m]
0
50
100
150
200
250
0 5 10 15 20 25
de
ad lo
ad g
[kN
∙m‾
¹]
span length L [m]
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50 60
de
ad lo
ad g
[kN
∙m‾
¹]
span length L [m]
0
10
20
30
40
50
60
70
80
0 10 20 30 40
de
ad lo
ad g
[kN
∙m‾¹
]
span length L [m]
0
10
20
30
40
50
60
70
80
0 5 10 15 20
de
ad lo
ad g
[kN
∙m‾¹
]
span length L [m]
0
10
20
30
40
50
60
70
80
15 20 25 30 35 40
de
ad lo
ad g
[kN
∙m‾
¹]
span length L [m]
51
To determine the reliability index and the probability of failure caused by the APT
load on a substandard bridge, it is necessary to evaluate the reliability of the
design code. In other words, the aim is to determine the reliability index and the
probability of failure of the bridge foreseen by the design code. The fundamental
hypothesis is that the designer has strictly followed the design code. It is assumed
that the probability of failure of the bridge that is implicit in the design code (PF0) is
the following (14):
50 10FP 5
0 10FP 50 10FP 5
0 10FP (14)
Consequently, the a priori reliability index (β0) is the following (15):
26410 510
10 .
FP (15)
where Φ is the normal cumulative distribution function with mean value μ=0 and
standard deviation σ=1.
In general the a priori reliability index can be expressed as follows (16):
20
20
0500500
SR
SR
%,%, (16)
where R50% and S50% are the mean values of the normal distributions of capacity
and demand respectively, σR and σS are the standard deviation of the normal
distributions of capacity and demand respectively, and the subscript 0 indicates
the design code.
The reliability index concerning the movement of the overweight load (βOL) can be
expressed as follows (17):
20
20
50050
SR
OLOL
SR
%,%, (17)
where S50%,OL is the mean value of the normal distributions of the demand caused
by the APT load.
The ratio between the two reliability indexes can be expressed as follows (18):
52
050050
50050
20
20
050050
20
20
50050
0 %,%,
%,%,
%,%,
%,%,
SR
SR
SR
SR
OL
SR
SR
OL
OL
(18)
If it is assumed that the ratio between the mean and the design value of the
demand caused by the overweight load (Sd,OL) is equal to the ratio between the
mean and the design value of the initial demand (Sd,0), we can write (19):
0
05050
,
%,
,
%,
dOLd
OL
S
S
S
S (19)
S50%,OL is expressed as follows (20):
10500
50050
0
5005050 %,
,
%,%,
,
%,%,%, S
R
SS
S
SSS
d
OL
d
OLOL (20)
where it is assumed that the design demand is equal to the design capacity
Sd0=Rd0. Equation (18) can therefore be expressed as follows (21):
1
1
1
0
0
050
050
050
050
050
050
050
050
0
m
mOL
S
S
S
R
S
S
S
R
%,
%,
%,
%,
%,
%,
%,
%,
(21)
where γm0 is the mean safety factor (22) (23).
050
095
095
0
0
0
0
05
05
050
050
0500
%,
%,
%,
,
,
,
,
%,
%,
%,
%,
%,
S
S
S
S
S
R
R
R
R
R
S
R d
d
d
dm (22)
53
R
SSRm
CoVk
CoVk
1
10 (23)
where γS and γR are the partial factor for the demand and capacity respectively,
and the relationships between the characteristic and mean value of the capacity
and demand are the following (24) (25):
RR CoVkRkRR 150505 %%% (24)
SS CoVkSkSS 1505095 %%% (25)
where R5% and S95% are the characteristic values of the normal distributions of
capacity and demand respectively, CoVR and CoVS are the coefficient of variation of
the normal distributions of capacity and demand respectively, and k is the
characteristic fractile factor equal to k=1.645.
Assuming the following values for the partial factor and for the coefficient of
variation of the demand and capacity, the relationship between the failure
probability and the lack of capacity is obtained (Fig. 24).
CoVR = 7.5%
CoVS = 15%
γR = 1.2
γS = 1.4
In Table 7 the association of the failure probability (Pf) with the lack of capacity (α)
of the bridge is shown.
54
Fig. 24: Relationship between failure probability and α
Table 7: Failure probability and lack of capacity
Failure probabilty Lack of capacity
Pf = 10-5
α=0
10-5
< Pf < 4∙10-5
0 < α < 0,1
4∙10-5
< Pf < 2∙10-4
0,1 < α < 0,2
2∙10-4
< Pf < 5∙10-4
0,2 < α < 0,3
Pf > 5∙10-4
α > 0,3
3.7 Conclusions
In order to limit costs, the assessment of the capacity of a bridge stock needs a
simplified and conservative approach first, and then, if a higher load rating is
required, a refining of the analysis. This Chapter has defined the assessment
procedure to manage the problem of the overweight traffic, with various simplified
levels of refinement, from Level 0 to Level 3. It has been shown that in the case of
free road traffic, the result of the assessment depends only on the maximum span
and on the year of bridge design (i.e. on the design loads of the design code).
Furthermore it is completely independent of the mechanical conditions of the
bridge. Moreover we obtained that in the case of a road closed to free traffic and
bridge transit in the center of the roadway, transit of the overweight load on the
bridge also depends on the transverse load distribution. The procedure has been
split for girder and arch bridges because of different resistance mechanisms of
the two typologies.
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
0 0.2 0.4 0.6 0.8 1 1.2
Fail
ure
Pro
bab
ilit
y
Lack of capacity α
55
To quantify the extent of the deficiency of the substandard bridges, a
classification has been made by defining the lack of capacity factor α. This index
reflects bridge understrength with respect to the overweight load and has been
defined as the percentage of the additional capacity needed to safely carry the
overweight load. Finally, the risk interpretation of parameter α in terms of the
probability of failure of a substandard bridge caused by the APT load has been
provided.
56
57
4 APPLICATION TO CASE STUDIES
4.1 Introduction
In Chapter 3, the general method to verify bridges under overweight loads was
shown. The assessment procedure involves three levels of refinement to improve
the verification of the bridge. In this Chapter I apply the methodology to two case
studies representative of the two typologies of bridges (girder and arch bridges) in
order to test the method. Although Level 3 assessment was not considered in the
case studies due to the costs involved in obtaining the updated material
properties based on in-situ tests, a simulation of this assessment level is
performed in order to show the methodology when in-situ tests are available.
4.2 Case studies
The two case studies analyzed are:
Fiume Adige bridge, SP90 km 1.159;
Rio Cavallo bridge, SS12 km 362.164.
As discussed above these two case studies are representative of the two
typologies in which bridges can be classified. Fiume Adige is a girder bridge while
Rio Cavallo is an arch bridge. It is worth pointing out that the methodology
proposed is valid for all the typologies of bridges, which are clearly more than two
(suspension bridges, cable-stayed bridges etc.). All types of bridges, except arch
bridges, can be evaluated as they were girder bridges. In fact, the method
foresees the comparison among demands caused by design loads and reference
loads (APT loads). Therefore, as discussed in Section 3.4.1, when the original
design load produces static stresses higher than the new overweight vehicle in a
58
simply supported span, the design stresses will be higher with a statically
indeterminate boundary condition. This is to say that the typology of the bridge is
irrelevant to the comparison (except arch bridges), so a simply supported
condition, choosing an appropriately span length, can always be assumed. For
instance in the case of a cable-stayed bridge the length among two cables must
be chosen for the calculation rather than the total span length, because the
cables are actually supports.
4.2.1 Fiume Adige
The bridge is located in the Municipality of Rovereto, close to the town of Trento,
Italy. The bridge is shown in Fig. 25. It is a four span prestressed concrete bridge,
built in 1980, with the maximum span L=33.12m. The deck has an overall width of
9.50m and carries a 7.50m roadway and two 1.25m sidewalks. The deck structure
of the 3 main spans is made up of 10 precast 1.44m double T girder elements per
span, with a 0.18m cast-in-place concrete deck, while the shortest span is made
up of 42 prefabricated elements pulled together with a cast-in-place concrete
deck.
The main dimensions of the structure are reported in the following:
bridge length: 106 m
first span
o length: 6.5 m:
o girder spacing and number of girders: 0.21 m / 42 elements
o total thickness of precast reinforced concrete slab: 0.45m
other spans
o length: 32.13m-33.12m
o girder spacing and number of girders: 0.92 m / 10 elements
o deck thickness: 0.18m
deck width: 9.5 m
carriageway width: 7.0 m
The design codes used by the bridge designer are the following:
D.M. 26/03/1980 “Norme tecniche per l’esecuzione delle opere in
cemento armato normale, precompresso e per le strutture metalliche.”
D.M. 02/08/1980 “Criteri generali e prescrizioni tecniche per la
progettazione, esecuzione e collaudo di ponti stradali.”
59
CNR-UNI 10018/72 “Istruzioni per il calcolo e l’impiego di apparecchi di
appoggio in gomma nelle costruzioni.”
a)
b)
Fig. 25: a) and b): “Adige Bridge”, Municipality of Rovereto (TN, Italy)
60
Fig. 26: Geometry of the Fiume Adige bridge
The lack of capacity () depends on the overweight load. Table 8 shows the lack
of capacity for each APT load using Level 0 assessment. These values are
calculated by a Matlab program (see Chapter 6) with the methodology described
in Chapter 3.
Table 8: Results of Level 0 assessment
APT load Level 0 assessment ()
56 tons 0.00
5x13 tons 0.14
6x12 tons 0.16
6x13 tons 0.21
7x13 tons 0.27
8x13 tons 0.33
9x13 tons 0.39
The design loads used by the bridge designer are the following:
permanent load of the deck (sum of structural and non-structural loads):
o G = 14.457t/m.
traffic loads:
32.12
2.7
0
8.5
2
9.2
26.6
76.5
6
6.4
2
106.13
32.6333.1332.687.15
1.00
1.601.60
1.60
1.80
6.44
1.00
31.68
0.60
1.05
31.63
1.00
C2 C3 C4 C5C1
Adige river
<--- p=2,1%I2I1I3 I4
P2 P3
S2S1
P1Isera
Rovereto
B
B
A
A
LONGITUDINAL SECTION
1.257.00
1.259.50
0.3
3
0.5
0
6.4
4
6.5
7
9.38
S1
1.257.00
1.259.50
0.5
0
2.0
2
9.10
7.00
0.50 0.508.00
P2
A-A TRANSVERSAL SECTION B-B TRANSVERSAL SECTION
Adige river
61
o lane number 1: uniformly distributed loads q1a = 4.509 t/m;
o lane number 2: uniformly distributed loads q1b = 2.166 t/m;
o remaining area: uniformly distributed loads qf=0.4t/m2.
the dynamic response coefficient of span 2, 3 and 4 is the following:
o Φ= 1.211
The design loads and the dynamic response coefficient reported above refer only
to spans 2, 3 and 4. This is because, for the sake of brevity, in the following I
report the details of the calculations only for these spans.
Level 1 assessment
As mentioned in Section 3.3, in Level 1 assessment the evaluator is required to
examine the design documents to verify whether the critical structural members
are oversized. This level of assessment is based only on the analysis of the
existing documents, without revising the assumed material properties or
calculation model. The evaluator repeats the structural calculations, replacing the
design traffic lane with the APT load. He/she calculates the effects caused by the
new loads and verifies that they are less than the capacity of the bridge. If one
verification is negative, the evaluator calculates the lack of capacity of the bridge
(α), as reported in Section 3.5. The verifications of the following structural
elements are performed:
decks;
columns;
abutments;
supports made of neoprene.
For each element a number of verifications are performed, such as the verification
of the bending moment and shear stress. For the sake of brevity, I report the
details of the calculations of the deck element only.
Free Travel
The design documentation includes the Massonnet-Guyon-Bares transverse load
distribution. In the following it is reported the calculation of the most stressed
beam that has an eccentricity of 1.410m from the center of the deck. In Fig. 27
the transverse position of loads and resulting Massonnet-Guyon-Bares transverse
load distribution are represented, while in Table 9 their values are reported.
62
Table 9: Transverse redistribution of loads for free travel
Loads Eccentricity (m) K (Massonnet) Q (t/m)
qf1 (Right side
remaining area) 4.000 1.254 0.4
q1a 1.750 1.563 4.509
q1b -1.750 0.761 2.166
qf2 (Left side
remaining area) -4.000 0.096 0.4
The total load on the most stressed beam can be calculated as follows (26):
[( ) ]
(26)
where ngirders is the number of prefabricated double T girder elements, equal to
ngirders=10.
The design value of the bending moment (Me) is the following (27):
(27)
where l is the span length.
The resistance moment reported in the design documentation is as follows (28):
(28)
Fig. 27: Transverse position of loads and resulting Massonnet-Guyon-Bares transverse
load distribution (free travel)
63
If the q1a load is replaced with, for example the 6 x 13 t = 78 tons APT load (Fig.
27 and Fig. 28), the design bending moment on the most stressed beam is equal
to (29):
(29)
Fig. 28: Transverse position of loads and resulting Massonnet-Guyon-Bares transverse
load distribution obtained replacing q1a with APT load (free travel)
Table 10 shows the verifications of the bending moment on the most stressed
beam for each APT load. The resistance moment reported was calculated by the
bridge designer for the design documentation. The safety factor foreseen by the
design code is equal to ηr =1.85, and then the verification is positive if the ratio
between the resistance moment and design moment is greater than:
(30)
Table 10: Bending moment verification (free travel)
APT load Mr (tm) Me (tm) ηr lack of capacity (α)
56t 620.56 294.49 2.107 0.00
5x13t 620.56 325.78 1.905 0.00
6x12t 620.56 330.19 1.879 0.00
6x13t 620.56 338.20 1.835 0.01
7x13t 620.56 350.62 1.770 0.05
64
8x13t 620.56 361.93 1.715 0.08
9x13t 620.56 373.21 1.663 0.11
The same procedure was followed to perform the shear verfication of the most
stressed beam. The shear verification involves comparing the shear stress
caused by the design load with the limit shear stress τc1=24 kg/cm2(D.M.
26/03/1980). The shear stress is equal to the sum of two contributions: the first
caused by the structural permanent loads considering only the prefabricated
double T girder elements and the second caused by the non-structural permanent
and traffic loads considering the whole cross-section (girder element + cast-in-
place concrete deck). The first contribution is independent of the traffic loads and
had been already calculated by the bridge designer. Therefore, only the second
contribution had to be calculated replacing the design load with the APT load.
For the sake of brevity the results of the verification are reported without showing
the calculations (Table 11).
Table 11: Shear verification (free travel)
APT load total shear stress τe
(Kg/cm²)
limit shear stress τc1
(Kg/cm²)
lack of capacity
(α)
56t 17.26 24 0
5x13t 17.91 24 0
6x12t 17.98 24 0
6x13t 18.12 24 0
7x13t 18.32 24 0
8x13t 18.51 24 0
9x13t 18.69 24 0
In addition to the ultimate verification here, I report also the serviceability
verification which involves checking the stress at the top (σS,TOT) and bottom (σI,TOT)
edge of the beam. The limits of compression (σc) and tension stress (σt) included
in the design code are the following:
σc < 0.38 Rck = 0.38∙460 kg/cm² = 174.8 kg/cm²
σt < 0.06 Rck = 0.06∙460 kg/cm² = (-) 27.6 kg/cm²
Also in this case the total stress is obtained by the sum of two contributions: the
first caused by the structural permanent loads considering only the girder
elements and the second caused by the non-structural permanent and traffic
loads considering the whole cross-section. Fig. 29 shows a part of the original
65
design documentation where the red arrow indicates the effects of the traffic load
that has to be replaced by the APT load. Table 12 shows the results of the
verification.
Fig. 29: scan of a part of the original design documentation
Table 12: Serviceability verification (free travel)
APT
load
σS,TOT
(kg/cm²)
lack of capacity
(α)
σI,TOT
(kg/cm²)
lack of capacity
(α)
56t 162.19 0.00 -2.60 0.00
5x13t 172.04 0.00 -20.69 0.00
6x12t 173.43 0.00 -23.25 0.00
6x13t 175.95 0.01 -27.88 0.01
7x13t 179.86 0.05 -35.06 0.04
8x13t 183.43 0.08 -41.61 0.07
9x13t 186.98 0.12 -48.13 0.11
The verifications are summarized in Table 13.
66
Table 13: Summary of Level 1 assessment (free travel)
APT load Level 0
()
Bending moment
Level 1
()
Shear stress
Level 1
()
serviceability verification
Level 1
()
56t 0.00 0.00 0.00 0.00
5x13t 0.14 0.00 0.00 0.00
6x12t 0.16 0.00 0.00 0.00
6x13t 0.21 0.01 0.00 0.01
7x13t 0.27 0.05 0.00 0.05
8x13t 0.33 0.08 0.00 0.08
9x13t 0.39 0.12 0.00 0.12
Bridge crossed in the center of the roadway
The verifications shown above have to be performed for traffic crossing the bridge
in the center of the roadway as well. The differences with respect to the
verification with free travel are the transverse position of the vehicle in the
carriageway and the absence of the second traffic lane.
The following shows the calculation of the most stressed beam, which, in this
case, is located in the center of the deck. Fig. 30 shows the transverse position of
loads and resulting Massonnet-Guyon-Bares transverse load distribution while in
Table 9 their values are reported.
Table 14: Transverse redistribution of loads for bridge crossed in the center of the roadway
Loads Eccentricity (m) K (Massonnet) Q (t/m)
Right side remaining area 4.000 0.748 0.4
APT load 0.000 1.467 var
Left side remaining area -4.000 0.378 0.4
67
Fig. 30: Transverse position of loads and resulting Massonnet-Guyon-Bares transverse
load distribution (bridge crossed in the center of the roadway)
For the sake of brevity, only the final results of the bending moment (Table 15)
and the serviceability (Table 16) verification are reported.
Table 15: Bending moment verification (bridge crossed in the center of the roadway)
APT load Mr (tm) Me (tm) ηr lack of
capacity (α)
56t 620.56 262.81 2.36 0.00
5x13t 620.56 292.17 2.12 0.00
6x12t 620.56 296.32 2.09 0.00
6x13t 620.56 303.83 2.04 0.00
7x13t 620.56 315.49 1.97 0.00
8x13t 620.56 326.10 1.90 0.00
9x13t 620.56 336.69 1.85 0.00
Table 16: Serviceability verification (bridge crossed in the center of the roadway)
APT
load
σS,TOT
(kg/cm²)
lack of capacity
(α)
σI,TOT
(kg/cm²)
lack of capacity
(α)
56t 152.20 0.00 15.72 0.00
5x13t 161.46 0.00 -1.26 0.00
6x12t 162.76 0.00 -3.66 0.00
6x13t 165.13 0.00 -8.01 0.00
7x13t 168.80 0.00 -14.75 0.00
8x13t 172.15 0.00 -20.89 0.00
9x13t 174.48 0.00 -27.01 0.00
68
As can be noticed in Table 15 and Table 16, the verifications are positive for each
APT load. The shear verification is not reported as it was already verified for free
travel.
Level 2 assessment
Level 2 assessments are needed when the verification at Level 1 is not satisfied.
In this case only the bending moment verification is required at Level 2 as the
shear verification was positive at Level 1.
As mentioned in Section 3.3, at Level 2 the evaluator reassesses the bridge
capacity using more refined models than those used by the original designer.
Often, a capacity increase is achieved by considering inelastic behavior and
spatial stress redistribution. In this case the deck is made by 10 simple supported
beams and the increase in the capacity is made possible by considering a
different transverse redistribution of the loads. In fact, the bridge designer used an
elastic transverse distribution of the loads and so it is possible to increase the
capacity by considering a plastic transverse distribution (Fig. 31).
Fig. 31: Plastic and elastic transverse distribution.
The value of qneg (Fig. 31) is equal to qneg = 0 in favor of safety, the unknowns are
x and qpos, which can be determined by imposing the equilibrium of the system.
The maximum uniformly distributed load qpos,max on the deck is equal to (33):
69
[
⁄ ]
(31)
where Mr is the resistance moment of the beam, 1.85 is the partial safety factor
and i =0.95 is the spacing between two girders.
Imposing the equilibrium of the system (32) and (33):
( ) Φ (32)
(
) (
) Φ (33)
where b = 9.50m is the width of the deck, ϕ = 1.211 is the dynamic response
coefficient, qf = 0.4 t/m, qneg=0, q1b = 2.166t/m, and G = 14.457 t/m is the sum of
structural and non-structural permanent loads.
Here, in the equilibrium of the system equations the loads were imposed.
Therefore, it is necessary to determine the APT uniformly distributed load (qAPT)
that produces the same effects (in terms of maximum bending moment) as the
real APT load (for example 6x13 tons APT load). If the 6x13 tons APT load is
considered, the 6x13t APT uniformly distributed load is equal to qAPT,6x13t = 4.813
t/m, and the following results are obtained:
x=9.07 m
qpos,6x13t=2.589 t/m
where qpos,6x13t is the resultant value of qpos considering the 6x13t APT uniformly
distributed load.
The verification is positive (see equation (34)) and the new partial safety factor is
obtained from equation (30):
(
)
(34)
Table 17 and Table 18 report the final results for each APT load and assessment
level.
70
Table 17: Summary of the final results (free travel)
APT load Level 0
()
Bending
moment
Level 1
()
Bending
moment
Level 2
()
Shear
stress
Level 1
()
Serviceability
verification
Level 1
()
56t 0.00 0.00 0.00 0.00 0.00
5x13t 0.14 0.00 0.00 0.00 0.00
6x12t 0.16 0.00 0.00 0.00 0.00
6x13t 0.21 0.01 0.00 0.00 0.01
7x13t 0.27 0.05 0.00 0.00 0.05
8x13t 0.33 0.08 0.03 0.00 0.08
9x13t 0.39 0.12 0.06 0.00 0.12
Table 18:Summary of the final results (bridge crossed in the center of the roadway)
APT load Level 0
()
Bending
moment
Level 1
()
Shear stress
Level 1
()
Serviceability
verification
Level 1
()
56t 0.00 0.00 0.00 0.00
5x13t 0.00 0.00 0.00 0.00
6x12t 0.00 0.00 0.00 0.00
6x13t 0.00 0.00 0.00 0.00
7x13t 0.02 0.00 0.00 0.00
8x13t 0.05 0.00 0.00 0.00
9x13t 0.08 0.00 0.00 0.00
Level 3 assessment
Level 3 assessment foresees that the evaluator can update the characteristic
values of the variables used in the assessment, based on the results of material
testing and observations. As discussed above, the procedure leaves the evaluator
free to test the materials, without specifying the minimum number of samples or
the type of test. However, if the evaluator wants to use the test results
quantitatively, a Bayesian probabilistic update technique should be applied.
In this case study, Level 3 was not considered due to the costs involved in
obtaining the updated material properties based on in-situ tests. However, a
simulation of Level 3 assessment has been performed in order to show the
methodology when in-situ tests are available.
71
Bayesian inference is a tool that is able to treat various types of information
following a unified approach (Sonda et al., 2003). In our case Bayesian inference
can be used to update the a priori probability distribution function using
information obtained from experimental tests (a posteriori) In Section 5.6 a brief
introduction to Bayesian statistics is given while in the following only the
application to the updating of a random variable is described (Fig. 32).
Fig. 32: Updating of a random variable (Sonda et al., 2003)
The basic assumption in the following is that all the random variables could be
described by Gaussian likelihood functions.
The a priori information of the random variable x can be expressed as follows
(35):
data collection through quality control Bayesian analysis
Updating of prior
information
a priori distribution
a posteriori distribution
72
(35)
where μ0 is the mean value and σ0 is the standard deviation of the a priori
Gaussian likelihood function. This distribution take into account all the
uncertainties that can be estimated a priori, including those regarding the mean
value. If μ1 represents the local uncertainties of the mean value μ0, μ1 can also be
considered a random value (36):
(36)
As can be noticed, the mean value is still equal to μ0, while the standard deviation
σ2 is a part of σ0, the standard deviation of the random variable x.
If μ1 is known, the conditional probability of the random variable x can be
expressed as follows (37):
(37)
where (38):
(38)
Where a sample of n observations for the random variable x is available (x ),
using Bayes’ theorem the conditional probability of μ1 can be expressed as
follows (39):
(39)
where is the likelihood and is equal to (40):
∫
(40)
Having determined from equation (39), the updated probability distribution
function of the random variable x, is equal to (41):
73
∫
(41)
From equations (35), (36) and (37), the following equation can be obtained (42):
(42)
where:
(43)
∑
(44)
(45)
The distribution function (42) is equal to equation (36) updated with the sample of
n observations of the random variable x. The parameters μ3 and σ4 are the
corresponding μ0 and σ2 assumed a priori.
Substituting equations (36) and (37) into (40), the updated distribution function of
the random variable x can be obtained (46):
(46)
where (47):
(47)
The methodology discussed above can be used here in the Fiume Adige case
study to update the properties of the materials. Level 3 assessment should be
used only in the case of a negative verification of Level 2 assessment. In this
case the bending moment verification provides negative results. The material that
most influences the capacity of the resistance moment of a precast girder is the
steel, as the concrete determines only the height of the neutral axis and so the
74
improvement in the capacity would be very small. Therefore, it is useful to perform
experimental tests only for the strands.
Let us suppose that the evaluator samples three strands and tests them with a
tensile standard test. The characteristic value of strength of prestressing steel
obtained from the design documentation is ftk=1900 MPa. Let us suppose that the
results of the tests are those reported in Table 19.
Table 19: Results of tensile standard tests of the strands
Sample ftk (MPa)
1 2021
2 1999
3 2219
We use the methodology discussed above and assume a Gaussian distribution
function for both a priori and a posteriori distribution of strength of prestressed
steel. We assume the following values that characterize the a priori distribution,
as recommended by Zandonini et al. (2012):
fptk=1900 MPa
σ0=74 MPa
which corresponds to a mean value of fpt0=2021 MPa. Where there is no
information on the value to assume for σ1 and σ2 Zandonini et al. (2012) advises
choosing (48):
(48)
From equations (38) and (48) we obtain (49):
(49)
Substituting the values in the equations (43), (44), (45) and (47) we obtain (50),
(51), (52) and (53):
(50)
75
∑
(51)
(52)
(53)
The updated characteristic value of strength of prestressing steel can be obtained
from the equation (54):
(54)
It can be noticed that the increment in the nominal value that can be used for the
calculations is equal to 3.6%. This means that in this case, Level 3 assessment
shows a decrease in the lack of capacity of 3.6%.
4.2.2 Rio Cavallo
The bridge is located in the Municipality of Calliano, close to the town of Trento,
Italy. The bridge is shown in Fig. 33. This is a four multiple arch bridge, built in
1940, with the span length l=12m. The carriageway is 11m in overall width. The
arch thickness at the crown is 0.60m and at the abutment is 0.90m. The
maximum thickness of the backfill is 1.9m.
The main dimensions of the structure are reported in the following and in Fig. 34:
Total bridge length: 55.4 m
Number of span: 4
Arch length: 12 m
Rise: 2.4 m
Arch thickness at the crown: 0.6 m
Arch thickness at the abutment: 0.9 m
deck width: 11 m
76
a)
b)
Fig. 33: a) and b): “Rio Cavallo”, Municipality of Calliano (TN, Italy)
77
Fig. 34: Geometry of the Rio Cavallo bridge
Table 8 shows the lack of capacity for each APT load using Level 0 assessment.
As for Fiume Adige bridge, these values are calculated by a Matlab program with
the procedure described in Chapter 3.
Table 20: Results of Level 0 assessment of the Rio Cavallo bridge
APT load Level 0 assessment ()
56t 0.20
5x13 tons 0.33
6x12 tons 0.30
6x13 tons 0.33
7x13 tons 0.33
8x13 tons 0.33
9x13 tons 0.33
The design code used by the bridge designer is the following:
MM.LL.PP. Normale n.8 15/09/1933
The bridge designer did not use the traffic loads foreseen by the design codes but
imposed a distributed uniformly load of 1000 kg/m2. He/she justified this choice
saying that for the arch bridge this load has greater effect than the design load.
2,6
2,6
2,6
55,4
1,62 11 1,62
2,95
2,6
0,8
2,95
A-A TRANSVERSAL SECTION
122,6
2,2
12 12 12
2,2
2,95
2,95
12 2.4
0,6
0,9
S1
A1
P1 P2 P3S2
A4A3A2
A
APERSPECTIVE DRAWING
LONGITUDINAL SECTION
78
The permanent loads used by the bridge designer are the following:
Arch weight: G1 = 2100 kg/m2
Backfill weight: G2 = 2100 kg/m2
Level 1 assessment
The verification of the arch is carried out with an equilibrium analysis checking the
most stressed cross-sections of the arch. This is the same procedure followed by
the bridge designer. Level 1 assessment is based only on the analysis of the
existing documents, without revising the assumed material properties or
calculation model. Although the calculation model is reported in the
documentation, it is impossible to simply replace the APT load with design load as
the verifications are obtained graphically. Since the load distributions are different,
the thrust line is different too. Automatically Level 1 assessment does not involve
improvements in the capacity and it is also necessary to make some additional
hypotheses to those made by the bridge designer. This means that Level 1
assessment is not useful for arch bridges and only Level 2 assessment can be
performed to verify the bridge.
Level 2 assessment
The equilibrium analysis made by the bridge designer foresees the verification of
the arch taking into account the loads acting on 1m of the arch width. For this
reason the verifications of the arch taking into account the free travel or the bridge
crossed in the center of the carriageway are the same. Assuming the same
hypothesis, here only half of the axle weight of the APT load is considered. In fact
the spacing between the two wheels is defined equal to 2m. This means that the
concentrated force acting on an arch of 1m width is equal to half of the axle
(13t/2=6.5t). Moreover, for the equilibrium analysis, the spread of the
concentrated forces of the APT load in the backfill is neglected. In other words, in
favor of safety, the load is considered as concentrated forces acting on the arch.
The bridge designer verified the arch with the following combination of the loads:
1) dead load (Fig. 35) that is equal to G1+ G2;
2) dead load and traffic load Q on the whole arch length (Fig. 36a);
3) dead load and traffic load Q on half arch length (Fig. 36b).
79
Fig. 35: Dead load
a) b)
Fig. 36: a) Dead load and traffic load on the whole arch length, b) dead load and traffic load
on half arch length
The same load combinations are considered, replacing the traffic load with the
APT load. Clearly the first combination has not been performed in the following,
as it remains the same as that computed by the bridge designer.
Firstly, the verification considering the APT load located on the whole arch length
is performed. Only the verification of the 9x13t APT load is reported here. It
should be noticed that if the 9x13t APT load is verified the other APT loads are
verified too.
To verify the arch element it is sufficient demonstrate that there exist a
satisfactory line of thrust that balances the given loading (Heyman, 1982). In other
words it is sufficient to find a thrust line in equilibrium with the external loading
which lies everywhere within the arch ring (Heyman, 1982).
To perform the equilibrium analysis of the arch, it is possible to study only half of
the arch length, considering the symmetry of the arch. As already mentioned, the
g
q
g
q
g
q
80
concentrated load is equal to half of the weight of one axle and only 1m width of
the arch is studied. Therefore, the load is equal to (55):
(55)
At the crown the load acting is equal to half of Q since only half of the arch is
studied. The numerical results and the equilibrium analyses (Fig. 37) are reported
in the following:
Fig. 37: Construction of the funicular polygon
Hs = 85351 kg
Hc = 60023 kg
where R is the resultant force of the loads, and Hs and Hc are the thrust at the
abutment and at the crown respectively. As can be seen in Fig. 37, the thrust line
is always within the middle third and consequently the cross-sections are always
compressed. Following the same procedure reported by the bridge designer, the
Hc
Hs
81
compression stresses (σI,c) at the crown and at the abutment (σI,s) are checked.
The results are reported in the following:
(56)
(57)
where A is the area, I is the momentum of inertia, y is the distance between the
baricenter and the compressive edge, and the subscripts c and s represent the
cross-sections at the crown and abutment respectively.
The compression stresses are very small and therefore the verification is positive.
Secondly, the verification considering the APT load located on half of the arch
length is performed. In this case the thrust line is not symmetrical and so the
whole arch is studied. The same hypotheses are assumed as in the previous
analyses. The numerical results and the equilibrium analysis (Fig. 38) are
reported in the following:
Hs = 83206 kg
Hc = 54181 kg
Fig. 38: Resulting thrust line with traffic load on half arch length
1
2
3 4
5 67
8 9 10 11 12 1314
15
16
17
18
82
As can be seen in Fig. 37, the thrust line is not always within the middle third and
consequently tension stress appears at the edge of some cross-sections. The
thrust line, however, is within the thickness of the arch and consequently the
equilibrium is guaranteed (Heyman, 1982). The results of the two most stressed
cross-sections (called section 8 and 14 for their position in the arch) and the
cross-sections at the crown and at the most unfavorable abutment are reported:
(58)
(59)
(60)
(61)
(62)
(63)
83
(64)
(65)
where the subscripts 8 and 14 represent the sections 8 and 14 respectively.
The results show that the stresses in the arch are very small and therefore the
verifications are positive. Also the shear stresses in the cross-sections of the arch
are always low and therefore the verifications are satisfied.
For the sake of brevity the calculations of the verifications of the piers and
abutments are not reported here. However, the total loads acting on these
elements are lower than the design load and so the verifications are automatically
positive.
In the following (Table 21 and Table 22) the final results for each APT load and
assessment level are reported. As can be noticed, the verification is satisfied in
both travel conditions (free travel and bridge crossed in the center of the
roadway).
Table 21: Results of Level 0 assessment (free travel)
APT load Level 0
assessment ()
Level 2
assessment arch
verification
()
Level 2
assessment pier
and abutment
verification
()
56t 0.20 0.00 0.00
5x13 tons 0.33 0.00 0.00
6x12 tons 0.30 0.00 0.00
6x13 tons 0.33 0.00 0.00
7x13 tons 0.33 0.00 0.00
8x13 tons 0.33 0.00 0.00
9x13 tons 0.33 0.00 0.00
84
Table 22: Results of Level 0 assessment (bridge crossed in the center of the roadway)
APT load Level 0
assessment ()
Level 2
assessment arch
verification
()
Level 2
assessment pier
and abutment
verification
()
56t 0.20 0.00 0.00
5x13 tons 0.33 0.00 0.00
6x12 tons 0.30 0.00 0.00
6x13 tons 0.33 0.00 0.00
7x13 tons 0.33 0.00 0.00
8x13 tons 0.33 0.00 0.00
9x13 tons 0.33 0.00 0.00
4.3 Discussion of the results
In Section 4.2 the results of the application of assessment Levels 1 and 2 to the
two case studies have been reported. For the girder bridge Fiume Adige, both
assessment levels have been applied while for the arch bridge Rio Cavallo only
Level 2 assessment has been performed because, as discussed above, there is
no sense in the application of Level 1 assessment for this type of bridge.
The results obtained for the Fiume Adige show that the precision of Level 1
assessment reveals a higher level of capacity than that given by the simplified
model in Level 0 assessment (i.e. verify potential reserve of capacity by
examining the design documentation). In fact, in the case of free travel and of the
9x13t APT load, the decrease in parameter α is 27% (see Table 23). This value
represents the overstrength of the bridge with respect to the design loads. The
reason of the large decrease is that there are many more reinforcing bars than
necessary. The critical verification at Level 0 assessment was the shear
verification but the design code involves a minimum transverse reinforcement
greater than necessary. It is worth emphasizing that this result could not be
obtained a priori as the extent of the excess reinforcing bars is not known. For the
limit state where Level 1 assessment provided negative results (i.e. the capacity
is still insufficient), Level 2 assessment was performed. Refining the models used
by the bridge designer, an additional decrease in the lack of capacity () has
been obtained. For the bending moment a plastic transverse distribution of the
load instead of elastic has been used. The additional decrease from Level 1 to
Level 2 assessment in the lack of capacity was of 6% (Table 23). For shear, more
85
refined models have not been used as the shear verification is positive in Level 1
assessment.
Table 23: Results of the case study Fiume Adige
Free Travel Level 0
()
Level 1
()
Level 2
()
APT load
9x13t 0.39 0.12 0.06
It should be noticed that the decrease in the lack of capacity between Level 1 and
Level 2 assessment is possible only for the limit states where an ultimate analysis
or a modification of the static model is possible. In general the serviceability
verifications cannot be considered in Level 2 as the elastic hypotheses of the
bridge designer cannot be changed.
The results obtained for the Rio Cavallo show that the bridge can withstand the
new APT loads. The verifications implied a complete re-assessment of the
elements of the bridge (arch, piers and abutments). All the verifications were
performed graphically drawing the funicular polygon in the elements. Safety is
calculated by checking that the thrust line is always within the elements and the
stress in each cross section is low enough. As can be seen in Table 24 the
verifications for the 9x13=117t APT load are positive. The large decrease in the
lack of capacity can be explained by the significant intrinsic geometrical factor of
safety of arch bridges. This is guaranteed by the high value of permanent loads of
arch bridges and by the hypothesis of the design. The assumption that the
material has an infinite compressive strength (Heyman, 1982) implies that the
stress are very low and in general an increase in the live load does not cause
structural problems to the bridge.
Table 24: Results of the case study Rio Cavallo
Free Travel Level 0
()
Level 2
()
APT load
9x13t 0.33 0.00
In both case studies, for the travel condition of the bridge being crossed in the
center of the roadway, the bridges are acceptable for each APT overweight
vehicle model. The reason is that the deck width is sufficient as there is more than
one line load and consequently that the vehicle crossing the bridge alone entails
the neglect of one or more line loads in the verifications.
86
4.4 Conclusions
In this Chapter Level 1 and 2 assessment of the girder bridge Fiume Adige and
the arch bridge Rio Cavallo have been applied in order to test the method. The
Chapter has shown how to apply the procedure to two real cases and the
necessity to apply the method to each design verification performed by the bridge
designer.
In the case of the girder bridge Fiume Adige, the results have shown that,
performing Level 1 assessment, all the APT loads less than or equal to the 6x12t
are acceptable, while performing Level 2 less than or equal to the 7x13t
(considering only the collapse verifications). Level 0 analysis had shown that only
the 56t APT load was acceptable for the bridge. A simulation of Level 3
assessment was shown using Bayesian probabilistic technique to update the
characteristic values of the variables used in the assessment, based on the
results of material testing and observations.
In the case of the arch bridge Rio Cavallo the results have shown that all the APT
loads, in both travel conditions, can cross the bridge without reducing the original
design safety level. Also in this case Level 0 analysis had shown that only the 56t
APT load was acceptable for the bridge.
In conclusion, Level 1 and 2 assessment can lead to good decreases in terms of
the lack of capacity of the bridge that were, for example, equal to 33% in the case
of 9x13t APT load (free travel condition, Fiume Adige bridge).
Once again, these results could not be obtained a priori as the overstrength of the
bridge and the design structural details are not known.
87
5 DECISION MAKING
5.1 Introduction
Decision making under uncertainty is about making choices whose consequences
are not completely predictable, because events will happen in the future that will
affect the consequences of actions taken now (Parmigiani & Inoue, 2009).
In Chapters 3 and 4 the criteria for re-assessing bridges for overweight loads and
the application to two case studies were shown. The application of Level 1 and 2
assessment were made automatically, without considering whether it was the
most rational choice. In fact, if the objective is to allow the travel of an overweight
load without reducing the original design safety level, when Level 0 assessment
shows that the lack of capacity is very high, the most rational choice could be the
full formal re-assessment of the bridge made by a structural engineer with
possible retrofit strengthening rather than proceeding with assessment levels 1, 2
and 3.
The aim of this Chapter is to define a decisional model to support the manager in
his/her choices. Therefore, the objective is to obtain a methodology to determine
the limit values of the lack of capacity (α), which make it more cost effective to
employ a certain assessment level (Level 1, 2 or 3) or a full formal re-assessment
with possible retrofit strengthening. In other words the goal is to propose a
decision making process to guide the manager in making the most cost effective
decision when selecting the appropriate assessment level.
In particular, in Section 5.2 and 5.3, I report the basic concepts of decision theory
and of expected utility theory. In Section 5.4 I define a general decision tree for
girder and arch bridges, and in Section 5.5 I show how to optimize the thresholds
of the lack of capacity α in order to minimize the expected costs. This aim is
pursued in order to guide the decision regarding the overweight load made by the
transportation agencies. In addition, in Section 5.6, a methodology to update
88
parameters on the basis of a number of case studies is proposed. Bayesian
statistics is used to implement this methodology.
5.2 Basic concepts
The symbology used in this Chapter is taken from the master’s thesis of Cappello
(2013), who studied in depth some of the aspects included in the present thesis
by applying expected utility theory with the aim of demonstrating its advantages.
The decision tree (Fig. 39) is the appropriate representation of any process
involving decisions. In Fig. 39 the action a1 starts from the decision node (a
square), and leads to the chance node (a circle), which in turn leads to two
possible consequences θ1 or θ2. The probabilities associated with the states θ1
and θ2 are the values of pθ1 and pθ2 respectively. zθ1|a1 and zθ2|a1 are the
outcomes observed after the occurrence of the situations θ1 and θ2, and u(zθ1|a1)
and u(zθ2|a1) are the utilities assigned to these outcomes.
In a general decision process it is possible to have many actions aj and the set of
these actions is called A. Similarly, the set of the states θi is represented by the
symbol Θ. One can imagine that the outcome zθi|aj corresponding to the condition
θi, observed after the action aj is chosen, is often, but not always, an amount of
money. The set of all the outcomes is called Z.
Once the utility function u(zθi|aj) is known, the utility associated with an outcome
zθi|aj can be calculated. The utility function represents the utility that a decision
maker wants to associate to an outcome. There can be several utility functions
and the result, i.e. the value of the utility, depends on which one is chosen.
Different functions u(zθi|aj) can be useful to represent different cases of decision
making process.
Fig. 39: Example of a decision tree and symbols used
Pθ1 θ1
a1
zθ1|a1, u(θ1|a1)
Pθ2 θ2
zθ2|a1, u(θ2|a1)
Decision node Chance node
89
5.3 Expected utility theory
The expected utility theory is a quantitative evaluation of the utility of the
outcomes. It provides the probability that each outcome will occur (Jordaan,
2005). The action chosen is the one that has the greatest expected utility.
The expected utility assigned to an action can be considered as the sum of the
utility of the outcomes (Cappello, 2013). The expected value can be calculated as
follows (66).
∑
(66)
where the probability of the outcome p(z), if the action a is chosen, is equal to
(67):
∫
(67)
As the higher U(a) the better, the optimal action a*, also called Bayes action, is
the one that maximizes U(a) (68):
(68)
which corresponds to the maximum expected utility (Parmigiani & Inoue, 2009).
If the Bayes action is chosen using losses, it is necessary to define the loss
function as the opposite of the utility (69):
(69)
where the space (θ,a) (Θ,A). Unlike the utility, the definition of the loss function
predicts that at least one action should be with zero loss (Cappello, 2013). As a
consequence, the regret loss function L(θ,a) has been defined (70), obtained as a
transformation of the utility-derived loss function (Cappello, 2013).
[ ] (70)
Defining the expected loss as follows:
90
(a) ∑
(71)
the Bayes action a* can be determined as follows (72), using the same principle
of the expected utility already applied (68):
(72)
5.4 Decision tree
In this Chapter, I develop a method that can support the decision-making process
for the APT bridge stock. In particular, the aim is to obtain the limit values of the
lack of capacity (α) which makes it more convenient to employ a certain
assessment level. The various simplified assessment levels that are available for
the APT, as a consequence of a negative verification, were shown in Table 4.
Firstly, the hypothesis is to use a single level tree diagram (Fig. 40), that is, to
assume that when an assessment level has been taken, the only possible results
are either the positive verification of the bridge or its replacement. The decision
tree in Fig. 40 is valid only for girder bridges, as applying Level 1 and 3
assessment to arch bridges is not useful (see Chapter 3 and 4). The methodology
explained in the following is only a simple example to introduce the problem of the
determination of the limit values of the lack of capacity. After this example the
final decision tree and a more rigorous discussion will be shown.
Fig. 40: Example of tree diagram
>t3
t2<≤t3
t1<≤t2
0<≤t1
0Level 0
Level 2
Replacement
Accept
0
1
2
3
Replacement
Accept
Replacement
Accept
Replacement
≤0
>0
Accept
Level 1
Level 3
91
Table 25 shows the cost of each type of assessment level as a function of the
total bridge cost (Cc). These are plausible values chosen by the author only in
order to explain the method. In the final decision tree the details of costs will be
shown and adapted to the real needs of the APT (see Chapter 6).
Table 25: Building cost percentage of different assessment levels.
Assessment Level Cost
Level 1 0.5%Cc
Level 2 1.5%Cc
Level 3 10%Cc
Replacement 100%Cc
A basic principle of the expected utility theory is here used to determine the
appropriate assessment level. Assuming a lognormal probability distribution for
the decrease in the lack of capacity from one assessment level to another, the
inverse cumulative distribution function describes the probability that, as a result
of a given assessment level, the verification becomes positive (73).
meanα
αln
β
1Φ1P (73)
where β is the lognormal standard deviation that incorporates aspects of
uncertainty; and αmean is the decrease in the lack of capacity after using a certain
assessment level. We imposed β equal to 0.7.
Table 26 lists the values of αmean that correspond to a probability of 50% that the
verification becomes positive. These values are the decrease in the lack of
capacity after the assessment level (see equation (74)). The values were chosen
by the author a priori. The mean decrease between two consecutive assessment
levels (Δα) is reported in Table 27.
(74)
Table 26: Coefficient corresponding to different assessment levels.
Assessment level αmean (αi)
Level 1 0.08
Level 2 0.12
Level 3 0.20
Replacement ∞
92
Table 27: Mean decrease between two consecutive assessment levels.
Assessment level Δαmean (Δαi)
Level 1 0.08
Level 2 0.04
Level 3 0.08
Replacement ∞
Fig. 41 shows the inverse cumulative distribution function obtained using the
values shown in Table 26. It represents the probability of obtaining a positive
verification after using a certain assessment level as a function of the lack of
capacity (α).
Fig. 41: Probability of obtaining a positive verification after using a certain assessment level
as a function of α.
Once the probability corresponding to a positive verification using a given
assessment level is obtained, the next step is to calculate the loss function. In this
case the loss function is equal to the cost, which depends on both the
assessment level and the consequence, according to equations (75) and (76).
( ) (75)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6
Pro
ba
bil
ity
α
Level 1 assessment
Level 2 assessment
Level 3 assessment
0.08 0.12 0.20
93
( ) (76)
The expected loss is equal to the following (77).
( ) (77)
Using equation (77), the relations as a function of the bridge cost Cc (Fig. 42) can be built.
Fig. 42: Relationship between costs and lack of capacity
The lower curve, given a value of α, is the Bayes action, that which has the lower
expected loss. Looking at the graph, it is therefore possible to determine the
intervals in which it is more appropriate to choose a given assessment level.
According to equation (77) it can be noted that there is a fixed cost (depending on
the assessment level (Table 25)) and a variable cost that depends on the
probability of obtaining a positive verification. The following values are obtained
(Fig. 42):
0%<α<2% Level 1 assessment is most cost effective;
2%≤α<6% Level 2 assessment is most cost effective;
6%≤α<47% Level 3 assessment is most cost effective;
α≥47% it is most cost effective to replace the bridge.
0%
20%
40%
60%
80%
100%
0 0.1 0.2 0.3 0.4 0.5 0.6
Co
stto
tal(%
Cc)
α
Level 1 assessment
Level 2 assessment
Level 3 assessment
0.02 0.06 0.47
1
3
2
94
As discussed above, in this first step the hypothesis is to use a single level tree
diagram, that is, to assume that when an assessment level has been applied, the
only possible results are the positive verification of the bridge or its replacement.
In reality it is possible that after an assessment level, the choice is to perform the
subsequent assessment level. Moreover, according to the APT (see Chapter 6),
there is no possibility of the bridge being replaced after Level 1. Replacement is
possible only after Level 2 assessment as the hypothesis is that if the result of
Level 1 assessment is negative, the only option is to perform Level 2 assessment.
Furthermore, it is not possible to perform Level 3 assessment directly without
having analyzed Level 1 and 2. These remarks are valid only for girder bridges as
for arch bridges only Level 2 assessment is possible.
The possibility of utilizing a higher assessment level, in the case of a negative
verification of a girder bridge, is illustrated in Fig. 43. The final decision tree for
girder bridges (Fig. 43) has four decision node. The possible actions are:
aa,(i): at Level i the APT load can cross the bridge, as can the overweight
loads with lower total gross weight;
ar,(i): at Level i the bridge should be replaced;
an,(i): the subsequent assessment level should be performed from Level i
to Level i+1;
while the possible outcomes are:
θs: the bridge withstands the overweight load;
θf: the bridge collapses;
θr: the bridge is replaced and the new bridge withstands the overweight
load.
95
Fig. 43: Final decision tree for girder bridges with documentation
L 1
L 0
L 2
L 3
αa(0
)
θs
θf
zθ
f|αa
(0), L
(zθ
f|αa
(0))
zθ
s|αa
(0), L
(zθ
s|αa
(0))
pθ
f|αa
(0)
1-p
θf|α
a(0
)
αr(0
)
zθ
r|αa
(0), L
(zθ
r|αa
(0))
θr
αr(2
)
zθ
r|αa
(2), L
(zθ
r|αa
(2))
θr
αr(3
)
zθ
r|αa
(3), L
(zθ
r|αa
(3))
θr
αa(1
)
θs
θf
zθ
f|αa
(1), L
(zθ
f|αa
(1))
zθ
s|αa
(1), L
(zθ
s|αa
(1))
pθ
f|αa
(1)
1-p
θf|α
a(1
)
αn
(0)
αn
(1)
αn
(2)
αa(2
)
θs
θf
zθ
f|αa
(2), L
(zθ
f|αa
(2))
zθ
s|αa
(2), L
(zθ
s|αa
(2))
pθ
f|αa
(2)
1-p
θf|α
a(2
)
αa(3
)
θs
θf
zθ
s|αa
(3), L
(zθ
s|αa
(3))
pθ
f|αa
(3)
1-p
θf|α
a(3
)
96
In Fig. 43 is the probability that the bridge collapses.
The symbols adopted are those used in a thesis by Cappello (2013), which as
mentioned before, studied in depth some of the aspects included in this thesis.
Cappello (2013) also studied the loss functions and the costs associated to the
possible collapse of bridges. In the decision tree of Fig. 43 the outcomes take into
account the possible collapse of the bridge after the decision to allow the crossing
of the bridge.
Unlike the decision tree of Fig. 40, in this case the expected loss as a
consequence of a given choice takes into account not only the cost of the
assessment level but also the consequences of the collapse of the bridge.
After Level 0 assessment (L0) the decision maker has three options:
aa,(0): the APT load can cross the bridge and the possible outcomes are:
o the bridge collapses;
o the bridge withstands the overweight load;
ar,(0): the bridge is replaced and the new bridge withstands the overweight
load;
an,(0): Level 1 assessment is performed and the result is α1.
After Level 1 assessment (L1) the decision maker has two options:
aa,(1): the APT load can cross the bridge and the possible outcomes are:
o the bridge collapses;
o the bridge withstands the overweight load;
an,(1): Level 2 assessment is performed and the result is α2.
After Level 2 assessment (L2) the decision maker has three options:
aa,(2): the APT load can cross the bridge and the possible outcomes are:
o the bridge collapses;
o the bridge withstands the overweight load;
ar,(2): the bridge is replaced and the new bridge withstands the overweight
load;
an,(2): Level 3 assessment is performed and the result is α3.
After the final level of assessment (L3) the decision maker has two options:
aa,(3): the APT load can cross the bridge and the possible outcomes are:
97
o the bridge collapses;
o the bridge withstands the overweight load;
ar,(3): the bridge is replaced and the new bridge withstands the overweight
load.
In the final decision tree for girder bridges, of Fig. 43, it has been supposed that
the design documentation of the bridge exists. Although it should always exist, in
many cases it does not (see Chapter 6). The decision tree needs to be modified
when the design documentation is not available. Without design documentation
Level 1 and Level 2 assessment are not possible, while Level 3 assessment can
be done but with additional cost with respect to Level 3 with documentation.
Firstly, the evaluator does not know the geometry of the bridge and a detailed
survey on the bridge geometry is necessary. Secondly, the material properties are
also unknown, and an extended campaign of in-situ testing and observations
must be done. Finally, the construction date must be estimated and a simulation
of the design process is necessary.
The decision tree for girder bridges without documentation is illustrated in Fig. 44:
Fig. 44: Decision tree for girder bridges without documentation
The same symbols used in the decision tree for girder bridges with documentation
are adopted.
After Level 0 assessment (L0) the decision maker has three options:
L 0 L 3
αa(0)
θs θf
zθf|αa(0), L(zθf|αa(0))zθs|αa(0), L(zθs|αa(0))
pθf|αa(0)1-pθf|αa(0)
αr(0)
zθr|αa(0), L(zθr|αa(0))
θr αr(3)
zθr|αa(3), L(zθr|αa(3))
θr
αn(0)
αa(3)
θs θf
zθf|αa(3), L(zθf|αa(3))zθs|αa(3), L(zθs|αa(3))
pθf|αa(3)1-pθf|αa(3)
98
aa,(0): the APT load can cross the bridge and the possible outcomes are:
o the bridge collapses;
o the bridge withstands the overweight load;
ar,(0): the bridge is replaced and the new bridge withstands the overweight
load;
an,(0): Level 3 assessment is performed and the result is α3.
After Level 3 assessment (L3) the decision maker has two options:
aa,(3): the APT load can cross the bridge and the possible outcomes are:
o the bridge collapses;
o the bridge withstands the overweight load;
ar,(3): the bridge is replaced and the new bridge withstands the overweight
load.
As discussed above, Level 1 and 3 assessment applied to arch bridges are not
useful. The decision tree for arch bridges is composed of only Level 0 and 2
assessment. As for girder bridges, we have to take into account whether the
design documentation exists. The difference between performing the Level 2
assessment with or without design documentation lies in the costs. The cost of
Level 2 with and without documentation (henceforth called respectively Level 2A
and Level 2B) are reported in Table 28 and Table 29 respectively:
Table 28: Costs for Level 3 assessment for arch bridges with design documentation
Item Cost
First inspection with geometric survey, assessment of
cracking, report
0.05%Cc
Calculation report 0.20%Cc
Total 0.25%Cc
Table 29: Costs for Level 3 assessment for arch bridges without design documentation
Item Cost
First inspection with geometric survey, analysis of the
cracking state, report
0.05%Cc
Geometric survey of the arch 0.20%Cc
In-situ test to check thicknesses 0.15%Cc
Calculation report 0.30%Cc
Total 0.70%Cc
99
The final decision tree for arch bridges is shown in Fig. 45
Fig. 45: Decision tree for arch bridges with design documentation
Fig. 46: Decision tree for arch bridges without design documentation
L 2AL 0
αr(0)
αn(0)
αa(0)
zθr|αa(0), L(zθr|αa(0))
θr
θs θf
zθf|αa(0), L(zθf|αa(0))zθs|αa(0), L(zθs|αa(0))
pθf|αa(0)1-pθf|αa(0)
αa(2)
θs θf
zθf|αa(2), L(zθf|αa(2))
pθf|αa(2)1-pθf|αa(2)
αr(2)
zθr|αa(2), L(zθr|αa(2))
θr
zθs|αa(0), L(zθs|αa(0))
L 2BL 0
αr(0)
αn(0)
αa(0)
zθr|αa(0), L(zθr|αa(0))
θr
θs θf
zθf|αa(0), L(zθf|αa(0))zθs|αa(0), L(zθs|αa(0))
pθf|αa(0)1-pθf|αa(0)
αa(2)
θs θf
zθf|αa(2), L(zθf|αa(2))
pθf|αa(2)1-pθf|αa(2)
αr(2)
zθr|αa(2), L(zθr|αa(2))
θr
zθs|αa(0), L(zθs|αa(0))
100
5.5 Optimization of alpha thresholds
The optimization of alpha thresholds has already been studied by Cappello (2013)
who based his analyses and models on the present work. However, it is worth
reporting the methodology of his thesis, which studied in depth the decision
making part of this thesis. His thesis presents the results of alpha thresholds and
Bayesian updating of parameters. In this Section and in Section 5.6 only the main
results are reported, while a comprehensive discussion of these topics are
available in the thesis by Cappello (2013). As Cappello’s thesis regards only
girder bridges, this Section reports the discussion on this typology. Therefore, the
optimization of alpha thresholds proposed in this Section is found for girder
bridges although the methodology is valid also for arch bridges. The optimization
of parameters will be applied to the APT bridge stock in Chapter 6.
The aim is to obtain the values of the possible actions aa,(i), an,(i), and ar,(i) which
minimize the expected loss. To do this, it is firstly necessary to define the
distributions for each stochastic variable (Δα)(i) that represents the decrease in the
lack of capacity shown after selected assessment level. According to Section 5.4
a lognormal distribution function is used and the mean values that correspond to a
probability of 50% that the verification becomes positive are reported in Table 26.
Secondly, the loss function should be defined and two loss functions have been
chosen to model the perception loss of the decision maker. The loss functions
chosen are reported in the following:
Bernoulli’s loss function;
probability of failure as an outcome.
In Bernoulli’s loss function the utility function proposed is as follows (78):
(
) (78)
that corresponds to the following loss function (79):
(79)
remembering that (80):
[ ] (80)
101
where z is money, z0 is the maximum amount of money the APT is willing to spend
and c is a coefficient to calibrate based on boundary conditions (see Cappello,
2013).
In the case of probability of failure as an outcome, the hypothesis is that the
decision maker of the APT cannot accept a bridge collapse. In this case the loss
is +∞ if pf=1, which is the worst possible condition. Consequently, this means that
the probability of failure can be considered as an outcome. The expected loss is
reported in the following (81):
(
) (81)
where c is a constant in millions of euro.
The costs of Table 25 were recalibrated after the effective assignments of the
assessment levels to the evaluators (see Chapter 6). The new costs are reported
in Table 30.
Table 30: Costs for Level 1 + Level 2 assessment for girder bridges.
Item Cost
Level 1: First inspection with geometric survey, report 0.08%Cc
>750 €
Additional cost for Level 2 assessment +0.12%Cc
Level 1 + Level 2 0.20%Cc
>750 €
Table 31: Costs for Level 3 assessment for girder bridges with design documentation
Item Cost
First inspection with geometric survey, report 0.05%Cc
Sampling and tests on the materials involved 0.30%Cc
Integration of the calculation report 0.15%Cc
Total 0.50%Cc
Table 32: Costs for Level 3 assessment for girder bridges without design documentation
Item Cost
Complete and detailed geometry survey 0.30%Cc
Sampling of slab 0.15%Cc
Inspection of the reinforcement bars 0.30%Cc
Sampling of the concrete of the girders 0.45%Cc
102
Sampling of the steel strands of the girders 0.60%Cc
Integration of the calculation report 0.50%Cc
Total 2.30%Cc
The cost for Level 0 assessment is equal to zero as the analysis is done
automatically using information contained in the APT-BMS database (see Chapter
6).
The results of the optimization of alpha thresholds for a bridge with design
documentation, performed by Cappello (2013), are reported in Table 33.
According to the APT (see Chapter 6) Cappello (2013) fixed the value of aa,(0) at
aa,(0) =0.04.
Table 33: Results of alpha thresholds for decision tree for girder bridges with
documentation
Alpha thresholds Bernoulli’s loss function Probability of failure as
an outcome
aa,(0) 0.01 0.01
an,(0) 0.55 0.28
aa,(1) 0.23 0.17
aa,(2) 0.29 0.19
an,(2) 0.55 0.28
aa,(3) 0.52 0.25
For a bridge without design documentation the results are shown in Table 34:
Table 34: Results of alpha thresholds for decision tree for girder bridges without
documentation
Alpha thresholds Bernoulli’s loss function Probability of failure as
an outcome
aa,(0) 0.04 0.04
an,(0) 1.07 1.00
aa,(3) 0.48 0.33
The total costs obtained when applying the decision trees defined in Section 5.4
are equal to € 29,200,000 (Cappello (2013)). These costs also consider the re-
construction cost of the bridge when, after the re-assessment, the verification is
negative.
103
5.6 Bayesian updating of parameters
Bayesian statistics helps to solve problems that involves inductive logic (Fig. 47),
which is where the cause can be determined by observing its effects (Cappello,
2013). For instance, it provides the probability that a fair coin is used if, after ten
tosses, the number of heads is six (Sivia, 2006). In contrast, deductive logic
predicts only the effects of the cause. Fig. 47 represents the difference between
inductive and deductive logic.
Fig. 47: A schematic representation of deductive logic, and inductive logic (Sivia, 2006).
Bayesian statistics is able to update prior distribution of parameters on the basis
of new observations (Bolstad, 2007). In our case the use of Bayesian statistics is
useful in order to update the distributions of the stochastic variables Δα using the
results of a number of case studies.
The Bayes’ theorem (85), which is the single tool of Bayesian statistics (Bolstad,
2007), can be obtained as follows:
(82)
where Pr(x,y) is the probability of observing both x and y, Pr(y) is the probability of
observing y, and Pr(x|y) is the probability of observing x given y. Since (83)
(83)
104
the following equations can be written (84) (85):
(84)
(85)
where Pr(y|x) is the likelihood probability function, Pr(x) is the prior probability
function, Pr(y) is the evidence and Pr(x|y) is the posterior probability function.
As reported in Section 5.4 a lognormal distribution function has been used for
each stochastic variable (Δα)(i). This means that we can write the associated
normal distribution function with mean μln(Δα),i=ln[(Δα)(i)] and standard deviation
σln(Δα)i=0.40 as follows (86):
[ ]
√
{
[
]
} (86)
where σln(Δα)i=0.40 results from β =0.7 as imposed in Section 5.4. Equation (86)
represents the a priori distribution of (Δα)(i). According to Section 5.4 the standard
deviation σln(Δα)i is the same for all assessment levels, while the mean μln(Δα),i is
reported in Table 35.
Table 35: Mean decrease from one assessment level to another.
Assessment level Δαmean (Δαi) μln(Δα),i
Level 1 0.08 -2.526
Level 2 0.04 -3.219
Level 3 0.08 -2.526
If the bridge is without design documentation (Level 3 assessment is carried out
after Level 0 assessment), equation (86) is still valid but in this case the mean
value of μln(Δα),3 is equal to the sum of μln(Δα),i , which is μln(Δα),3=ln[0.08+0.04+0.08]=
ln[0.20]. The distributions of (Δα)(i) are shown in Fig. 48.
105
Fig. 48: Distributions of (Δα)(i)
The Bayes’ theorem (85) can be rewritten as follows (Cappello, 2013) (87):
(87)
where g is the posterior distribution without the constant Pr(y), fp is the prior
distribution, and fl is the likelihood distribution defined as follows (Cappello, 2013)
(88):
∏ ( [ ]
)
(88)
where equation (89)
( [ ]
) (89)
provides the probability density of a normal distribution with mean μ*ln(Δα),i and
standard deviation σ*ln(Δα),i at the point ln[(Δα)(i)]k.
The prior distribution according to equation (86) is reported in the following (90):
0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.4 0.5
pd
f[Δα
(i)]
Δα(i)
i=1,3
i=2
106
( | |)
( ( ) ( ) | ( )|) (90)
where the values of μln(Δα),i are presented in Table 26 and σln(Δα),i=0.4. As can be
noted, fp is a prior distribution of the parameters. The prior distribution is obtained
by multiplying two normal distributions as μ*ln(Δα),i and σ*
ln(Δα),i are independent
(Cappello, 2013). Cappello (2013) imposed the coefficients of variation as follows
(91) (92):
(91)
(92)
which represent respectively the coefficient of variation of the mean and the
coefficient of variation of the standard deviation of the prior distribution.
Solving equation (87) with a Markov Chain Monte Carlo (MCMC) method, it is
possible to obtain the updated values of Δα (μ*ln(Δα),i) and β (σ*
ln(Δα),i). MCMC
methods are numerical methods developed to obtain samples from the posterior
distribution. The Metropolis-Hastings algorithm, which will be used in Section 6.8
for the updating of parameters, is an MCMC method. The steps of Metropolis-
Hastings algorithm are reported below (Bolstad, 2010):
1) start with an initial value θ(0);
2) do from n=1 to n:
draw θ’ from q(θ(n-1),θ’);
calculate the probability α(θ(n-1),θ’);
draw u from U(0,1);
if u< α(θ(n-1),θ’) then let θ(n)= θ’, or else let θ(n)= θ(n-1).
where q(θ,θ’) is a candidate distribution that generates a candidate θ’ given a
starting value θ, U(0,1) is a uniform distribution, and α(θ(n-1),θ’) is the acceptance
probability that is equal to (93):
107
{
} (93)
where g(θ|y) is the posterior distribution.
5.7 Conclusions
The aim of this Chapter was to define a decisional model to support the manager
in his/her choices. Different decision trees have been defined for arch and girder
bridges with and without design documentation. The separation of decision trees
is necessary due to the different costs of the assessment levels with or without
design documentation and the different assessment methodologies for arch and
girder bridges. Furthermore, a methodology has been defined to determine the
limit values of the lack of capacity (α) that make it more cost effective to employ a
certain assessment.
In addition, a methodology has been proposed to optimize these thresholds of the
lack of capacity α, and a method to update these parameters when a number of
case studies are available. These methodologies will be applied to the APT bridge
stock in Chapter 6.
108
109
6 APPLICATION TO THE APT BRIDGE STOCK
6.1 Introduction
As discussed in Chapter 2, transportation agencies do not know how to respond
to transit applications on their bridges. In particular, The Department of
Transportation of the Autonomous Province of Trento adopts a conservative
procedure to release permits based on empirical information from past crossings
of loads that caused no apparent damage to the bridges. Chapter 3 illustrated a
methodology that takes into account the real capacity of the bridges. Moreover a
decisional model to support the manager in his/her choices was defined.
In this Chapter, I apply both the procedure defined in Chapter 3 and the
decisional model shown in Chapter 5 to the APT bridge stock.
In Section 6.2 I report a brief introduction to the APT’s Bridge Management
System (APT-BMS) while in Sections 6.3 I report the strategic objectives to
assess/retrofit the APT bridge stock to allow free or restricted travel of minimum
overweight load vehicles over the entire network. Then, in Section 6.4 the
consequent results obtained by applying Level 0 assessment are shown. Section
6.5 shows the final decision tree adopted by APT in order to guide their choices.
Based on this decision tree a prediction of re-assessment results and costs is
performed in Section 6.6.
In the last Sections the optimization of the thresholds of the lack of capacity
(Section 6.8) based on the results of 20 APT bridge case studies (Section 6.7) is
performed.
6.2 Bridge Management System
Since 1998, following a policy of decentralization that has relocated state
responsibilities to the provinces, the number of bridges managed by the
110
Department of Transportation of the Autonomous Province of Trento, a medium-
sized public agency, has increased from 412 to approximately 1000 (Bortot et al.,
2009).
As discussed in Section 1.1, in order to allow effective planning of maintenance
policies and to meet the new requirements, the Department of Transportation of
the APT has begun to collaborate with our Department of the University of Trento
(DIMS), to develop a comprehensive Bridge Management System (APT-BMS)
(Bortot et al., 2009).
The objective was to develop a management tool which was able to determine
present and future needs for maintenance, reinforcement and replacement of
APT bridges (Zonta et al., 2007). Moreover the aim was to provide a prioritization
system to guide the APT in the effective utilization of available funds. The result
has been the development of a management system fully available on the web
(www.bms.provincia.tn.it). Fig. 49 shows the main components and the
information flow of the APT-BMS. The system is based on a set of components:
database, cost, deterioration, repair and maintenance models, inspection data,
and decision making. Each component consists of a procedure package and is
divided into modules. The modules can operate on a project level (single bridge)
or network level (bridge stock) (Zonta et al., 2007).
The database consists of:
Inventory data;
Condition State (CS) data;
Reliability data.
Inventory data includes all the information and a simplified model of the bridge,
representing the logical distribution of its elementary units (Zonta et al., 2007).
111
Fig. 49: Main components and information flows of the APT-BMS (Zonta et al., 2007)
The model consists of structural units (SU) which in turn are divided into standard
elements (SE), most of which coincides with AASHTO CoRe elements (AASHTO,
1997) (Fig. 50). This division allows the definition of the Condition State evaluated
during inspections.
The data of the Condition State provide information about the deterioration of
standard elements and their intensity (Bortot et al., 2009). The condition state of
the bridge is detected by the evaluators, following the procedures of the APT-
BMS. The condition state is represented by an integer. The minimum value, which
represents the optimum situation, is equal to 1 while the maximum can vary to 3,
4 or 5 depending on the type of standard element. This methodology was defined
in order to maintain the same evaluation models used by one of the most
common management systems, PONTIS (Thompson, 1998) (Bortot et al, 2007).
112
Fig. 50: Example of bridge representation as implemented in the APT-BMS (Zonta et al.,
2007)
In addition to this heuristic condition index, another condition index, the Apparent
Age (AA) index, is introduced in the APT-BMS reflecting the specific view of the
agency on the concept of bridge deterioration. The AA index calculates the most
likely age of the bridge, given the condition state of its elements. The index,
compared to the real age, shows whether the bridge deterioration is normal, or if
the structure has suffered abnormal degradation (Zonta et al., 2008).
Moreover, the APT-BMS involves a specific section for the formal assessment of
the reliability of bridge structures. Reliability data directly concerns the capacity of
the bridge, and consists of a set of reliability indexes β, each associated with an
ultimate limit state and a specific structural unit or substructure. Further
information on reliability and the APT-BMS is given in Zonta et al., (2007); .
In the APT-BMS there are three types of inspections:
Inventory inspections;
Routine inspections;
Special inspections.
Inventory inspections collect all data required for the operation of the system and
if possible include attachments, such as pictures, design documents, FEM
models, and AutoCad files, while routine inspections are used to update the
condition state of the bridge. Routine inspections are divided into simplified and
principal inspections depending on the level of detail and the frequency of the
assessment. Special inspections are performed only when, a routine inspection
identifies structural anomalies, or the inspector was unable to evaluate one or
more elements of the bridge.
113
The prioritization approach adopted in the APT-BMS is based on the following
principle: priority is given to those actions that, given a certain budget, will
minimize the risk due to a number of unacceptable events in the whole network in
a specific timespan (for example: in the next tL=5 years) (Zonta et al., 2009).
The ranking of any intervention using a priority index can be expressed as
follows (94):
(94)
where RTOT(tL) is the total risk of unacceptable events over the time span (0,tL]
when the intervention is not undertaken, and C is the cost of bridge reconstruction
(Zonta et al., 2009).
The unacceptable events considered in the APT-BMS are:
failure of a principal element;
failure of a secondary element;
pile collapse due to scour;
accidents due to sub-standard guardrails;
loss of life due to an earthquake.
Assuming that the unacceptable events Ei are non-correlated, the total
cumulative-time risk can be evaluated as the sum of risks associated with each
event considered (95):
∑
∑
(95)
where Ri is the risk associated with the unacceptable event Ei, PE is the probability
of occurrence of the unacceptable event and PX|E is the magnitude of the expected
damage if the event occurs.
The system operates entirely on the web, and is based on an SQL database,
accessible over the Internet. The database can be modified through a web-based
user-friendly interface. Inspectors and evaluators can find the appropriate
procedures on the same web application, and can upload the results of condition
assessments or of safety evaluations. The manager can view the result of the
analysis on the same web-interface. The system is continuously updated by our
research group at the University of Trento.
114
The system is supported by a number of separate servers which are hosted on
different machines and can be managed by different actors (Zonta et al., 2006). In
its initial configuration the system includes (Fig. 51):
a database server;
a web server application;
the data analysis application server.
Fig. 51: Architecture of the APT-BMS in the single agency configuration (Zonta et al.,
2006).
The web application consists in a user-friendly portal that provides access for
data management to users. Users are:
system managers (full control on the stock information);
inspectors and evaluators (access limited to bridges and inspections
under their responsibility);
guests.
One of the most interesting aspects of the APT-BMS is its fully web-based
operation. The various site sections include: Inventory, Inspections, People,
Roads and Network.
The Inventory is the first section of the web application (Fig. 52) and includes a
basic search engine, a result list including the important characteristics of the
bridges, and a multi-tab window showing information on the selected bridge
(Zonta et al., 2006). Inventory data include all the information regarding the
bridge, namely identification, administrative issues, geographical location,
construction and previous retrofits, and also a number of multimedia attachments
(e.g. pictures, documents, FEM models, AutoCAD files). The tab II livello shows
DB
WEB
INTERFACE
ANALYSIS
APPLICATION
SERVER
MANAGER
WEB
INTERFACE
WEB
INTERFACE
WEB
APPLICATION
SERVER
INSPECTORS
GUESTS
SYSTEM
UPDATE
GROUP
115
the elementary model of the bridge, broken down into Standard Units, and
Standard Elements. Moreover, an advanced search engine and a complete report
generator are included in the section (Zonta et al., 2006).
Fig. 52: Display of the Inventory section (Zonta et al., 2006).
The Inspections section allows the manager to organize the inventory, routine
inspections, and safety evaluations (Fig. 53). The manager assigns tasks to each
inspector/evaluator, and validates the results uploaded by them at the end of their
job. In the case of an inspection that foresees a condition assessment, the
evaluator assigns a condition index to each Standard Element of the bridge.
When there are anomalies, the evaluator inserts a detailed description and a
number of digital pictures. For safety assessment, the evaluator fills in a table with
the results, where each limit state considered in the evaluation has an associated
load rating factor or a reliability index (Fig. 54). After entering data, the evaluator
should attach a detailed report of the calculations done, including all electronic
files used during the analysis (e.g. the input file of a Finite Element program).
inventory module
sample multimedia record
116
Fig. 53: Display of Inspections section (Zonta et al., 2006).
Fig. 54: Display of the evaluations section (Zonta et al., 2006).
In the People and Roads sections the manager can list the people involved in the
process (including the application users), and the APT road network, respectively.
In the Network section there are the results of the risk analysis of all the
unacceptable events of the bridge stock. The users can visualize the results on
Google Earth maps.
inspections management
inspection attachment
evaluator reports the results of the evaluation in a table
a detailed technical report is also attached
117
6.3 Strategic objective
The APT has defined a ten-year objective to assess/retrofit its bridge stock to
allow free or restricted travel of minimum overweight load vehicles over the entire
network. In particular, the road system has been divided into two different
networks: a strategic network and a non-strategic network. The strategic network
(see Table 36) includes all inter-regional highway links, with 574 km of highways,
264 girder bridges and 87 arch bridges; the non-strategic network is the
remaining 1836 km of local road links, including 428 girder bridges and 174 arch
bridges (Fig. 55).
Table 36: Strategic network
Links Start kilometer End kilometer
SP59 - -
SP71 4.000 40.000
SP72 - -
SP76 - -
SP79 10.200 13.200
SP80 - -
SP83 0.000 7.700
SP90 tronco1 3.200 14.000
SP90 tronco dir Ala - -
SP90 tronco2 - -
SP90 dir SS12 - -
SP90 dir Rover - -
SP90 tronco3 5.700 7.510
SP101 - -
SP104 - -
SP118 3.700 6.367
SP232 - -
SP232 dir - -
SP235 0.000 17.400
SP254 Rupe - -
SS12 326.181 350.300
357.300 383.000
387.300 401.275
118
SS12svincoli - -
SS42 147.846 170.900
SS43 0.000 30.650
SS45bis 112.000 116.800
119.100 154.160
SS47 73.014 130.8 (ex 131.800)
ex SS47 Martignano - -
SS48 16.430 20.400
35.000 63.800
SS48 var Predazzo - -
SS50 61.057 74.400
SS237 - -
SS239 - -
SS240 0.000 16.800
SS240 dir - -
SS249 96.500 101.130
SS347 0.000 11.780
Fig. 55: Girder and arch bridges on a)strategic and b)non-strategic network
The mid-term objective for the strategic network is to allow the free travel of 6-
axle overweight vehicles with a maximum axle weight of 12 tons (120 kN) and
total weight 72 tons (720 kN), and the restricted travel (bridge crossed in the
center of the roadway) of 8-axle overweight vehicles with a maximum axle weight
of 13 tons (130 kN) and a total weight of 104 tons (1040 kN). Similarly, for non-
264
87
girder bridges arch bridges
428
174
girder bridges arch bridges
a) b)
119
strategic bridges, the targets are the free travel of any 56 ton vehicles and the
restricted travel of 72 ton vehicles.
Table 37: Strategic objective
Free travel Center of roadway
Strategic network 6x12t=72t 8x13t=104t
Non-strategic network 56t 6x12t=72t
6.4 Level 0 assessment results
Level 0 assessment involves the verification of the bridge assuming no
overstrength. As already discussed in Chapter 3, in practice the assessment
consists in the calculation, for each section of the deck, of the maximum bending
moment and shear caused by the design load (or the eccentricity of the thrust line
in the case of arch bridges) and the subsequent comparison with the APT loads.
In the case of free travel, the APT-BMS database contains the required data to
perform the assessment of each bridge, which are the following:
maximum span length;
year of the bridge design (required to infer the design code).
The assessment in the case of crossing the bridge in the center of the roadway
would require the Massonet-Guyon-Bares coefficients to evaluate the transverse
distribution of the loads. As an alternative, all the geometrical and mechanical
details of the deck must be known. Currently, the APT-BMS database does not
contain this information although it has been planned to enter the values of the
Massonnet-Guyon-Bares coefficients for each bridge in the next two years. For
this reason, the transverse distribution of the loads has been calculated using the
Courbon method, which requires only the deck width as input data, which is
contained in the APT-BMS database.
The analysis of Level 0 assessment was done automatically using the information
contained in the APT-BMS database (see Chapter 6). A Matlab® program (Matlab
v. 7.0.1) was implemented which automatically reads the geometrical data of
each bridge and calculates, in each section of the bridge, the maximum bending
moment and the maximum shear stress caused by the design loads and the APT
loads. The design loads are calculated by inferring the design code from the year
of the design of the bridge. The following design codes are implemented in the
program:
120
Normale n.8 15/09/1933
Normale n.6018 09/06/1945 and n.772 12/06/1946
Circolare n.820 15/03/1952
Circolare n.384 14/02/1962
D.M. 02/08/1980
D.M. 04/05/1990
D.M. 14/01/2008
For bridges designed before 1933, the program implements the traffic loads
included in Jorini (1918), which reports the reference loads to be used in the
design of bridges.
Fig. 56 and Fig. 57 show the flowcharts implemented in the Matlab program in the
case of free travel and bridge crossed in the center of the carriageway
respectively. The details and a comprehensive discussion of the Matlab program
can be found in the master’s thesis of Lastei (2011).
Fig. 56: Flowchart of the Matlab program implemented for Level 0 assessment in the case
of free travel
Initialization
Uploading data from BMS database
(span length, design year, width of the deck,
typology of superstructure, type of network)
Uploading design loads
END
arch bridge?NOYES
Calculation of the envelope of
the eccenticity
Uploading APT loads
Uploading design loads
Calculation of the envelope of
bending moment and shear
stress
Uploading APT loads
Calculation of the envelope of
bending moment and shear
stress
Creation of Google Earth Maps
Comparison and calculation of the lack of
capacity
for i=1 to n_bridges
Calculation of the envelope of
the eccenticity
121
Fig. 57: Flowchart of the Matlab program implemented for Level 0 assessment in the case
of bridge crossed in the center of the carriageway
Fig. 58 shows the results of Level 0 assessment using the desired overweight
loads for both the strategic and the non-strategic network. Each dot on the map
corresponds to one bridge, while the dot color encodes the assessment result in
terms of the index with the following meanings:
Table 38: Lack of capacity and assigned colors
Lack of capacity Color
α =0
0 < α < 0.1
0.1 < α < 0.2
0.2 < α < 0.3
α > 0.3
Fig. 58a plots the distribution of the lack of capacity in the case of strategic
network of the bridges located in the APT, using the higher value of alpha
between that relative to free travel (with the 6x12t APT load) and that a relative to
Initialization
Uploading data from BMS database
(span length, design year, width of the deck,
typology of superstructure, type of network,
traffic divider)
END
traffic divider?NOYES
Creation of Google Earth Maps
for i=1 to n_bridges
Uploading design loads
calculation of the envelope of
bending moment and shear
stress (Courbon method)
Uploading APT loads
calculation of the envelope of
bending moment and shear
stress (Courbon method)
Uploading design loads
arch bridge?
NOYES
Calculation of the envelope of
the eccenticity
Uploading APT loads
Uploading design loads
calculation of the envelope of
bending moment and shear
stress (modified Courbon
method)
Uploading APT loads
calculation of the envelope of
bending moment and shear
stress (modified Courbon
method)
Calculation of the envelope of
the eccenticity
Comparison and calculation of the lack of
capacity
122
bridge crossed in the center of the roadway (with the 8x13t APT load). Fig. 58b
shows the results for the non-strategic network with the desired overweight loads.
a) b)
Fig. 58: deficiencies for target APT load models on a)strategic and b)non-strategic
networks (maps)
a) b)
Fig. 59: deficiencies for target APT load models on a)strategic and b)non-strategic
networks
The analysis of the strategic bridges shows that the verification is positive for 29%
(103/351 bridges) of the bridges of the APT stock. For these bridges, the APT can
authorize the transit of overweight loads up to 72 tons in the case of free travel,
and up to 104 tons in the case of bridge crossed in the center of the roadway. For
non-strategic bridges, Level 0 assessment is positive for 58% (347/602).
The remaining bridges (71% for strategic and 42% for non-strategic network)
have been classified as sub-standard, in order to define a priority ranking for
specific verification or for future reinforcement or replacement.
The results of assessment levels were inserted in the APT-BMS database
through three tables (Table 39, Table 40, and Table 41). In particular, the table
10369
98
16
65
0
50
100
150
200
250
300
350
400
α=0 0<α<0.1 0.1<α<0.2 0.2<α<0.3 α>0.3
347
5369
45
88
0
50
100
150
200
250
300
350
400
α=0 0<α<0.1 0.1<α<0.2 0.2<α<0.3 α>0.3
123
Transitabilità (Table 39) contains the results of all the assessments (Level 0,
Level 1, Level 2, Level 3) and from this table the data are uploaded to the web
application.
Table 39: Example of the new table Transitabilità implemented in the APT-BMS
Oid Oid_B Tip_APT
load
Trav_
Cond α
Tip_Design
_code Ass_L Lim_state Tip_Us Des
1 1 1 1 0 3 0 3 6 -
2 1 2 1 0.23 3 0 4 10 -
3 1 3 1 0.23 3 1 2 5 -
4 1 4 1 0.28 3 1 6 4 -
5 1 5 1 0.34 3 2 7 2 -
6 1 6 1 0.53 3 3 5 1 -
7 1 7 1 0.55 3 3 5 7 -
8 1 1 2 0 3 0 9 6 -
9 1 2 2 0 3 0 5 4 -
10 1 3 2 0 3 1 1 9 -
11 1 4 2 0 3 1 4 11 -
12 1 5 2 0.10 3 1 8 4 -
13 1 6 2 0.15 3 1 3 3 -
14 1 7 2 0.25 3 1 1 2 -
Oid is a progressive number, while Oid_B is the Oid number of the bridge.
Tip_APT_load, Trav_Cond, and α are the APT load, the travel condition, and the
lack of capacity corresponding to the assessment. Tip_Design_code is the design
code used in the evaluation, Ass_L is the assessment level used in the
evaluation, and Lim_state is the limit state that caused the highest lack of
capacity, Tip_Us is the element where this value of α is calculated, and Des is a
detailed description of the verification.
The results of Level 0 assessment are automatically implemented in the
database, while for the other assessment levels the evaluator can insert the data
directly from the web application.
The results of assessment levels for each bridge are shown in the web application
in the new Tab Transitabilità (Fig. 60).
124
Fig. 60: New tab Transitabilità (rio delle Seghe bridge) (www.bms.provincia.tn.it)
125
The other two tables were implemented separately from the table Transitabilità, in
order to ensure that it is possible to add other APT loads or design codes in the
future.
Table 40: New table APT loads implemented in the APT-BMS database
Oid APT_load
1 56t
2 13x5=65t
3 12x6=72t
4 13x6=78t
5 13x7=91t
6 13x8=104t
7 13x9=117t
Table 41: New table Design Code implemented in the APT-BMS database
Oid Design_code
1 DM_14/01/2008
2 DM_04/05/1990
3 NT_02/08/1980
4 Circ_1962
5 Circ_1952
6 Circ_1945-1946
7 Norm_1933
8 Ante_1933
6.5 Decision tree
In Section 5.4, a general decision tree was defined to guide the APT manager in
making the most cost effective decision when selecting the appropriate
assessment level and in Section 5.5, the values of α thresholds were found.
The APT decided not to perform any assessment levels if the value of α, after
whatever assessment level, is less than 4%. The reason is that even if the
verification is negative, the value of α is acceptably small and no further
assessment is required. This limit value was obtained by imposing the probability
of collapse caused by the APT load on a substandard bridge equal to twice the
probability of collapse foreseen by the design code. According to Section 3.6,
which shows the risk interpretation of parameter α, the value of the lack of
capacity that corresponds to twice the original probability of collapse is equal to
126
equation (97). This equation is obtained from equation (96), whose meaning is
explained in Section 3.6.
The APT decided to follow an heuristic method, rather than using the alpha
thresholds of Chapter 5. However, the values chosen are partly based on the first
example of a decision tree shown in Section 5.4.
Φ Φ( [(
) ]) (96)
( Φ
) ( Φ
) (97)
Fig. 61: Final APT decision tree for girder bridges
In the case of arch bridges the final APT decision tree is reported in Fig. 62. In
this case the alpha thresholds were also chosen by the APT following an heuristic
method.
L 1
OK OK OK OKα<0.04
Full formal
assessment
Full formal
assessment
Full formal
assessment
OK
Full formal
assessment
With design documentation
Without design documentation
L 0 L 2 L 3
L 3
α<0.04 α<0.04
α<0.04
α <0.04α <0α <0 α <0 α <0.04
α >0.40
α <0.04α <0
α <0.04α <0
α >0.47
127
Fig. 62: Final APT decision tree for arch bridges
6.6 Prediction of re-assessment results and costs
The aim of this Section is to predict the results and costs when the decision trees
shown in Section 6.5 are applied. The prediction is based on the lack of capacity
shown in the results of Level 0 assessment. Table 27 shows the decrease in α
from one assessment levels to another.
Fig. 63 outlines the decision-making scheme and shows the prediction of re-
assessment results for girder bridges. The results show both strategic and non-
strategic bridges. More specifically, after Level 0 analysis, 433 bridges are
automatically assessed, including 49 bridges with α<0.04, while 22 others are
unlikely to pass any further level of simplified assessment and so must be
formally reassessed and possibly strengthened. 139 bridges, out of the remaining
237, have no design documents and must be directly assessed under a Level 3
procedure. The other 98 bridges, with design documents, can undergo the multi-
level assessment procedure. After the reassessment, the APT estimates that
there will be 72 substandard bridges out of the 692 total bridges. These bridges
will be analyzed individually and possibly strengthened to resist higher loads.
L 2A
Full formal
assessment
OK
Full formal
assessment
With design documentation
Without design documentation
L 0
L2B
α<0.04
α <0.04α <0
OK
Full formal
assessment
α<0.04
α <0.04
OK α<0.04
α <0.04α <0 α <0
α > 1.51
128
Fig. 63: Anticipated results of re-assessment for girder bridges
Fig. 64 shows the prediction of re-assessment results for arch bridges. As for
girder bridges, the results show both the strategic and non-strategic networks.
Fig. 64: Anticipated results of re-assessment for arch bridges
After Level 0 analysis, 69 bridges are automatically assessed. Among the
remaining 192 bridges, 175 have no design documents; the other 17 bridges, with
design documents, can undergo the multi-level assessment procedure. After
L 1
OK
(384)
OK
(27)
OK
(39)
OK
(7)
α<0.04
(49)
Full formal
assessment
(22)
Full formal
assessment
(4)
Full formal
assessment
(6)
OK
(76)
Full formal
assessment
(40)
L 0 L 2 L 3
L 3
α<0.04
(14)
α<0.04
(1)
α<0.04
(23)
237692 1471
α <0
139
98
With design documentation
Without design documentation
L 2A
Full formal
assessment
(0)
OK
(165)
Full formal
assessment
(10)
With design documentation
Without design documentation
L 0
L2B
α<0.04
(0)
175
OK
(17)
Full formal
assessment
(0)
α<0.04
(0)OK
(66)
α<0.04
(3)
261 17192
129
reassessment, the APT estimates that there will be 10 substandard bridges out of
the 261 total.
Table 42 and Table 43 show the total costs for each assessment level for girder
and arch bridges respectively (see Chapter 5 for itemized costs). Level 3S
indicates the cost of Level 3 whether the design documentation is not available.
Table 42: Total cost of assessment levels for girder bridges
Action Cost
Level 1 0.08%Cc
Level 1 + Level 2 0.18%Cc
Level 3 0.50%Cc
Level 3S 2.30%Cc
Table 43: Total cost of assessment levels for arch bridges
Action Cost
Level 2A 0.25%Cc
Level 2B 0.70%Cc
These costs (Table 42 and Table 43) can be applied to estimate the total cost of
decision trees. Table 44 and Table 45 show the expected cost obtained by
applying decision tree for girder bridges and arch bridges respectively. The total
cost for girder bridges is equal to € 2,118,000 while for arch bridges it is €
699,000. It can be noted that there is a high number of bridges without
documentation, both for girder (139, which corresponds to 58%) and arch bridges
(175, which corresponds to 91%). In addition these numbers are calculated
among sub-standard bridges and so the number of bridges without design
documentation is higher. The main reason is the policy of decentralization that
has relocated state responsibilities to the provinces with a consequent transfer of
the design documentation that has not happened yet. For arch bridges, however,
the absence of the design documentation is due to the year of the design, which
in almost all cases is before the second world war, and in many cases the
documentation has been lost.
130
Table 44: Expected cost applying decision tree for girder bridges
Action No. bridges Total
(k€)
Total per bridge
(k€)
Level 1 27 36 1.33
Level 1 + Level 2 71 105 1.48
Level 3 14 47 3.36
Level 3S 139 1930 13.88
TOTAL
2118
Table 45: Expected cost applying decision tree for arch bridges
Action No. bridges Total
(k€)
Total per bridge
(k€)
Level 2A 17 27 1.59
Level 2B 175 672 3.84
TOTAL
699
Missing design documentation adds extra cost. In fact, the main part of the total
cost is due to Level 3S assessment (or Level 2B for arch bridges), which is, for
girder bridges, one order of magnitude higher than the cost of assessment levels
with design documentation. Although for arch bridges the difference is smaller,
the cost per bridge of Level 2B assessment (€ 3,840) is more than double that of
Level 2A assessment (€ 1,590). Table 46 and Table 47 show the comparison
between costs with and without documentation for girder and arch bridges
respectively. These estimated costs were obtained applying the decision tree of
Fig. 61 and Fig. 62 supposing that the design documentation of all bridges is
available. It can be noted that the saving by owning the design documentation of
all bridges would be equal to € 2,097,000 (€ 1,650,000 and € 447,000 for girder
and arch bridges respectively). In the case of girder bridges the value of a single
design documentation is € 11,800, and for arch bridges is € 2,600. This means
that it is worth focusing efforts to look for the design documentation of bridges.
Table 46: Comparison between costs with and without documentation for girder bridges
Girder bridges
Cost without documentation 2118 k€
Cost with documentation 468 k€
Saving 1650 k€
Number of bridges without documentation 139
Value of documentation 11.8 k€
131
Table 47: Comparison between costs with and without documentation for arch bridges
Arch bridges
Cost without documentation 699 k€
Cost with documentation 252 k€
Saving 447 k€
Number of bridges without documentation 175
Value of documentation 2.6 k€
Table 48 illustrates the total costs obtained by adding the costs for arch and
girder bridges. As in Section 5.5, these costs also consider the reconstruction
cost of the bridge when, after the re-assessment, the verification is negative.
Table 48: Total costs
Arch bridges Girder bridges Total
Re-assessment
cost
0.699 M€ 2.118 M€ 2.817 M€
Reconstruction
cost
8.610 M€ 61.990 M€ 70.6 M€
TOTAL 9.309 M€ 64.108 M€ 73.417 M€
6.7 Application to 20 APT bridges
In this Section the APT decision trees defined in Section 6.5 are applied to 20
APT bridges, using the same assessment procedure discussed in Chapter 3 and
showed in detail in the two case studies Fiume Adige and Rio Cavallo. The aim is
to verify the bridges with the APT loads defined in the strategic objectives of
Section 6.3. Therefore, as explained in Section 6.5, Level 2 assessment is
applied when the lack of capacity after Level 1 assessment is more than 4%.
Table 49 and Table 50 list, for strategic and non-strategic bridges respectively,
the APT substandard bridges selected for the assessments and show the lack of
capacity after Level 0 assessment.
132
Table 49: 8 strategic bridges selected for assessments
Bridge Identification Year Typology α
rio Secco SS12 km 364.745 1950 Precast reinforced concrete
slab 75%
Rio delle Pile SP 79 km 12.00 1984 Precast reinforced concrete
slab 30%
Rocchetta SS43 km 23.400
Dx 1950 Concrete arch 27%
rio Ala SS12 km 337.442 1937 Concrete arch 16%
Fiume Adige SP 59 km 0.255 1978 Precast reinforced concrete
girder 15%
Rio delle
Seghe SP 71 km 32.705 1969
Precast reinforced concrete
girder 12%
rio Silla SP 104 km 1.150 1982 Precast reinforced concrete
slab 8%
fiume Sarca SS 240 km 16.5 1961 Reinforced concrete girder 8%
Table 50: 12 non-strategic bridges selected for assessments
Bridge Identification Year Typology α
Fiume Adige SP 21 km 2.895 1943 Reinforced concrete
girder
84%
Torrente Maso SP 65 km 12.89 1966 Reinforced concrete
girder
33%
torrente Arione -
Aldeno
SP 25 km 0.525 1971 Precast reinforced
concrete girder
12%
rio Piazzina SP 31 km
35.191
1971 Precast reinforced
concrete girder
12%
rio delle Stue SP 31 km 31.63 1972 Precast reinforced
concrete girder
11%
rio Val della Setta a
Bondone
SP 69 km 9.19 1969 Reinforced concrete
girder
11%
rio Lenzi SP 135 km
11.920
1976 Precast reinforced
concrete girder
10%
rio Redebus SP 8 km 13.580 1973 Precast reinforced
concrete girder
10%
Torrente Grigno SP 78 km
13.310
1972 Precast reinforced
concrete girder
10%
133
rio Loner SP 89 km
20.420
1977 Precast reinforced
concrete slab
10%
torrente Arione-
Covelo
SP 25 km 3.49 1975 Precast reinforced
concrete girder
10%
Echernachel SP 8 km
14.320
1973 Precast reinforced
concrete girder
10%
Table 51 and Table 52 list the results of assessment for strategic and non-
strategic bridges respectively.
In a number of sets of design documentation the verification of piers and
abutments was not reported. In these cases it is possible to compare the design
and the APT load. This assessment represents an advanced Level 0 as, in
contrast with the normal Level 0, the permanent loads are known exactly.
Table 51: Results for strategic bridges
Bridge Year Typology α α_new Assessment
Level
rio Secco 1950 Precast reinforced
concrete slab 75% 3% Level 1
Rio delle Pile 1984 Precast reinforced
concrete slab 30% 0% Level 1
Rocchetta 1950 Concrete arch 27% 10% Level 2
rio Ala 1937 Concrete arch 16% 12% Level 2
Rio delle
Seghe 1969
Precast reinforced
concrete girder 12% 0% Level 2
Fiume Adige 1978 Precast reinforced
concrete girder 15% 1% Level 1
rio Silla 1982 Precast reinforced
concrete slab 8% 5% Level 2
fiume Sarca 1961 Reinforced concrete
girder 8% 8% Level 2
134
Table 52: Results for non-strategic bridges
Bridge Year Typology α α_new Assessment
Level
Torrente Maso 1966 Reinforced
concrete girder 33% 1% Level 1
Fiume Adige 1943 Reinforced
concrete girder 84% 47% Level 2
torrente Arione -
Aldeno 1971
Precast reinforced
concrete girder 12% 5% Level 1
rio Piazzina 1971 Precast reinforced
concrete girder 12% 0% Level 1
rio delle Stue 1972 Precast reinforced
concrete girder 11% 0% Level 1
rio Val della Setta
a Bondone 1969
Reinforced
concrete girder 11% 0% Level 1
rio Lenzi 1976 Precast reinforced
concrete girder 10% 0% Level 1
rio Redebus 1973 Precast reinforced
concrete girder 10% 0% Level 1
Torrente Grigno 1972 Precast reinforced
concrete girder 10% 2% Level 1
rio Loner 1977 Precast reinforced
concrete slab 10% 0% Level 2
torrente Arione-
Covelo 1975
Precast reinforced
concrete girder 10% 0% Level 1
Echernachel 1973 Precast reinforced
concrete girder 10% 2% Level 1
The results obtained show that 19/20 (95%) bridges have decreased and only
one bridge has maintained the lack of capacity estimated after Level 0 analysis.
The mean decrease in α was equal to 12% (16% for strategic network and 9% for
non-strategic network). For 7 bridges (Fig. 65), Level 2 assessment was
necessary and possible, while for 2 bridges it was not possible to complete Level
1 or Level 2 assessment because of the lack of the design documentation of piers
or abutments. For these elements only an advanced Level 0 has been done. In
the case of the non-strategic network, 7/12 bridges (58%) are safe, and 10/12
have a lack of capacity less than 4%. For the strategic network, 2/8 bridges (25%)
are formally verified and 4/8 (50%) have a lack of capacity that is less than 4%.
135
Fig. 65: Results of the application of the decision tree to 20 APT bridges
The results show that:
with the exception of particular cases, the strategic objectives in the case
of the non-strategic network are satisfied;
in the case of the strategic network the verification of strategic objectives
is more difficult and Level 2 assessments are often required;
in some cases Level 1 and Level 2 assessments are not very useful due
to the lack of the design documentation of piers or abutments, which in
many cases are the most unfavorable elements.
It is interesting to look at the decrease in the lack of capacity as a function of the
design year of the bridge (Fig. 66).
L 1
OK (7)
20 7L 0 L 2
α<0.04 (5)
no verification
of piers or
abutments
(1)
OK (2)
α<0.04 (0)
no verification
of piers or
abutments
(1)
α<0.08 (1)
α<0.14 (2)
α<0.47 (1)
α<0.08 (1)α<0.08 (1)
136
Fig. 66: Decrease in the lack of capacity as a function of the design year of the bridge
As can be noticed, the variability of the decrease in the lack of capacity is higher
for bridges designed before the 1960s. For bridges designed from 1960 onwards,
the mean decrease in the lack of capacity is 10%.
It is important to point out that in four cases the decrease in the lack of capacity is
very large (>30%). In the case of Rio Secco bridge (Fig. 67), Δα from Level 0 to
Level 1 assessment is equal to Δα=72%. The reason is that the Italian Army
Authority (Ispettorato Arma del Genio) imposed design loads (military load)
greater than those foreseen by the design code (civil load). The Rio Secco bridge
was built immediately after the Second World War (1950), and the Italian Army
Authority wanted to preserve its strategic networks.
0%
10%
20%
30%
40%
50%
60%
70%
80%
1930 1940 1950 1960 1970 1980 1990
Δα
design year of the bridge
137
Fig. 67: Rio Secco bridge (www.bms.provincia.tn.it)
As can be noticed in Fig. 67, two signposts placed by NATO at the beginning of
the bridge indicate the maximum loads that can cross the bridge in the case of
free travel and bridge crossed in the center of the roadway. If there are NATO
signposts, we can infer that the design loads are military. In conclusion, in these
cases the design load could be higher than that foreseen by the design code and
therefore we suggest that those particular cases should be analyzed in depth.
In the other three cases (Torrente Maso bridge, Rio delle Pile bridge, Fiume
Adige bridge), the reason for the large decrease in the lack of capacity is that
there was some incorrect data within the APT-BMS database. The inventory
inspector had classified the bridges as second category, which means the design
loads are lower than those effectively used, although they should have been
classified as first category.
This demonstrates that the automatic program used to perform Level 0
assessment cannot recognize the rough mistakes contained in the APT-BMS
database. This could become a significant problem if the bridge was classified as
first category but it was in reality a second category. The solution to this problem
could be investigated in future work.
The assessment results have been implemented in the web application of the
APT-BMS (www.bms.provincia.tn.it). Fig. 60 shows the summary of the results in
the web application (in the inventory section) of rio delle Seghe bridge for each
APT load, for both free travel and bridge crossed in the center of the carriageway.
138
This table is generated automatically by the web application, summarizing the
results inserted in the inspection section by the evaluator. In fact, he or she
completes a more detailed table that contains the results of all the verifications
performed for each transit condition, assessment level, structural element, and
limit state. In this manner the APT-BMS manager can quickly check the reason
for any lack of capacity summarized in the inventory section. In the APT-BMS
application can also be downloaded the assessment report of the evaluator.
It is important to highlight that even though the procedure requires only the
strategic objectives to be verified, the other APT loads should also be analyzed.
This means that in the same assessment level, the evaluator must analyze each
APT load, while the decision to apply the subsequent assessment level depends
only on the result of the APT load chosen in the strategic objectives. This is
because the cost of the assessment level is independent of the number of loads
to verify. The APT could authorize, in the case of positive verification, the crossing
of overweight loads higher than those defined in the strategic objectives.
6.8 Bayesian updating and thresholds of optimization
In Section 6.7, 20 APT bridges are analyzed and the decrease in the lack of
capacity from one assessment level to another is obtained. In this Section I use
these data to update the distribution of (Δα)(i) (see equation (86)) in order to
optimize the alpha thresholds (Section 6.8) to minimize the expected cost. Only
the results concerning girder bridges are used to update the values, as only the
thresholds for girder bridges are shown in the following. The methodology has
been already described in Section 5.6.
Table 53 shows the mean and the standard deviation of (Δα)(i) for each
assessment level after the evaluations.
Table 53: Mean and standard deviation of (Δα)(i) for each assessment level after the
evaluations
Assessment Level Δαmean (Δαi) σ(Δαi)
Level 1 0.16 0.17
Level 2 0.045 0.059
Applying the Bayes updating (87) the following results are obtained (Fig. 68 and
Table 54).
139
Fig. 68: Updated distributions of (Δα)(i)
Table 54: Updated distributions of (Δα)(i)
Assessment Level Δαmean (Δαi) σln(Δα)i
Level 1 0.076 1.27
Level 2 0.005 2.26
As can be noticed, the updated value of the mean in the case of Level 1
assessment is close to that a priori (which was equal to 0.08), while for Level 2
assessment it is less, (0.04). In contrast, the updated values of standard deviation
σln(Δα)i are both larger than those a priori. This means that the variability of Δαi is
large and consequently the probability of obtaining a large decrease is also high.
The shape of the lognormal distribution is wider for values greater than the mean
value. In conclusion, the updated mean values are smaller than those a priori (in
particular for Level 2 assessment), but the standard deviations are larger and so
we can expect a larger decrease in the lack of capacity when an assessment
level is performed.
Applying the same methodology and costs already shown in Section 5.5, the
results shown in Table 55 are obtained.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.5 1 1.5 2 2.5 3 3.5
pd
f[Δα
(i)]
Δα(i)
i=1
i=2
140
Table 55: Results of optimization of alpha thresholds (decision tree for girder bridges with
documentation)
Alpha thresholds Bernoulli’s loss function Probability of failure as
an outcome
αa,(0) 0.15 0.00
αn,(0) 0.55 0.28
αa,(1) 0.20 0.16
αa,(2) 0.29 0.19
αn,(2) 0.55 0.28
αa,(3) 0.52 0.25
The results are quite different using Bernoulli’s loss function or the probability of
failure as an outcome. The reason is that the loss functions are very different from
each other. As can be noticed, in both cases the thresholds αa,(i) increase during
the transition to the following assessment level, while αa,(i) is constant.
Fig. 69 shows the decision tree for girder bridges with documentation and the
results of optimization of alpha thresholds using Bernoulli’s loss function.
141
Fig. 69: Results of optimization of alpha thresholds using Bernoulli’s loss function (decision
tree for girder bridges with documentation)
L 1
L 0
L 2
L 3
αa(0
) < 0
.15
αr(0
) > 0
.55
θr
αr(2
)>
0.5
5θ
rα
r(3)
>0
.52
θr
αa(1
) <0
.20
αa(2
)<
0.2
9
L (α
a(0
),α(0
))
zθ
r|αa
(0), L
(zθ
r|αa
(0))
zθ
r|αa
(2), L
(zθ
r|αa
(2))
zθ
r|αa
(3), L
(zθ
r|αa
(3))
L (α
a(1
),α(1
))L
(αa
(2),α
(2))
L (α
a(3
),α(3
))
142
6.9 Conclusions
In this Chapter I apply the decisional model to the APT bridge stock after having
described the strategic objectives of the APT. The mid-term APT objective is to
assess in the next 10 years: the strategic road network for 72 ton free travel
vehicles, and 104 ton restricted travel vehicles; and the non-strategic network for
56 ton free travel vehicles, and 72 ton restricted travel vehicles. The final decision
tree adopted by the APT has been defined for reassessing existing bridges using
a multi-level verification procedure. Based on Level 0 analysis, of the 953 bridges
in the stock, 502 are automatically acceptable, while 429 need further re-
assessment before being formally acceptable. To re-assess substandard bridges,
the APT has launched a re-assessment program, expecting to find about 82
substandard bridges that need retrofit strengthening. The prediction of costs
shows that, with the current design documentation available, the total re-
assessment cost is equal to € 2,817,000. If the design documentation of all
bridges were available the total re-assessment cost would be equal to € 720,000
with a total saving equal to € 2,097,000. Hence it is very important to endeavor to
locate for the design documentation of bridges.
Finally, this Chapter has also described the optimization of the thresholds of the
lack of capacity based on the results of 20 APT bridge case studies. The results
show that as the number of assessment increases, the thresholds α can be
updated in order to optimize the choices of the decision maker. In comparison to
the a priori distribution, the updated probability distribution function shows that the
updated mean values are smaller than those a priori (in particular for Level 2
assessment), but the standard deviations are larger.
143
7 CONCLUSIONS AND FURTHER WORK
7.1 Conclusions
In this thesis I illustrated how the problem of overweight traffic management can
guide transportation agencies, focusing on the legal issues entailed by
overweight/oversize load permits. More specifically, practical criteria have defined
for issuing overweight load permits, based on the definition of a number of
reference load models and on two travel conditions (free and restricted traffic). It
has been demonstrated that in the case of free road traffic, the transit of the
overweight load on the bridge depends only on the maximum span and on the
year of bridge design. Furthermore it is completely independent of the mechanical
conditions of the bridge. Moreover it has been obtained that when of a road is
closed to free traffic and the bridge is crossed in the center of the roadway, transit
of the overweight load on the bridge also depends on the transversal load
distribution.
A protocol has been defined for reassessing existing bridges using a multi-level
verification procedure and, tested with two case studies. The results show that the
decrease in terms of the lack of capacity of the bridge can be great. In fact, for
both cases, it was equal to 33% in the case of a 9x13t APT load (free travel
condition).
I applied utility theory to optimize the expected cost in order to propose a process
to guide the APT in making most cost effective decision when selecting the
appropriate level of verification. The choices of the decision maker depend on the
lack of capacity of the bridge obtained from Level 0 analyses. Different decision
trees have been defined for arch and girder bridges with and without design
documentation. The separation of decision trees is necessary due to the different
costs of the assessment levels with or without design documentation and the
different assessment methodologies for arch and girder bridges. In addition, a
144
methodology has been proposed to optimize these thresholds of the lack of
capacity α, along with a method to update these parameters when a number of
case studies are available. These methodologies have been applied to the APT
bridge stock in accordance with the strategic objectives to assess/retrofit the APT
bridge stock. Based on the decision tree proposed and Level 0 analysis 502 of
the 953 bridges are automatically acceptable, while 451 need further re-
assessment. It is worth emphasizing that the APT is protected against legal
liability because the APT authorizes the transit of an overweight load that the
bridge designer would have allowed. The prediction of costs shows that, with the
current design documentation available, the total cost for reassessment of the
APT bridge stock is equal to € 2,817,000 (Table 56). If the maximum expected
utility principle was applied to determine the thresholds of the lack of capacity of
the decision trees instead of using an heuristic approach, the total saving would
be equal to 44,300,000 €, which corresponds to a 60% less.
Moreover, a methodology has been proposed for the optimization of the
thresholds of the lack of capacity based on a number of bridge case studies.
Table 56: Comparison between costs with and without documentation
Cost without documentation 2817 k€
Cost with documentation 720 k€
Saving 2097 k€
Number of bridges without documentation 314
Value of documentation 6.7 k€
7.2 Further work
The methodology proposed in this thesis has been applied to the APT stock using
information contained in the APT-BMS database. It was necessary to implement
the Courbon coefficient to obtain the transversal load distribution required to
calculate the lack of capacity in the case of bridge crossed in the center of the
carriageway. This was due to the absence of all the geometrical and mechanical
details of the deck in the APT-BMS database necessary to calculate the
Massonnet-Guyon-Bares coefficients. However, as it is planned to enter the data
of the Massonnet-Guyon-Bares coefficients for each bridge in the next two years,
it is recommended to re-assess Level 0 analyses using these more accurate
coefficients. Although the overall results should remain similar, the lack of
capacity of a number of particular cases could change.
145
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